:: The Properties of Product of Relational Structures
:: by Artur Korni{\l}owicz
::
:: Copyright (c) 1998-2021 Association of Mizar Users

registration
let S, T be non empty upper-bounded RelStr ;
cluster [:S,T:] -> upper-bounded ;
coherence
[:S,T:] is upper-bounded
proof end;
end;

registration
let S, T be non empty lower-bounded RelStr ;
cluster [:S,T:] -> lower-bounded ;
coherence
[:S,T:] is lower-bounded
proof end;
end;

theorem :: YELLOW10:1
for S, T being non empty RelStr st [:S,T:] is upper-bounded holds
( S is upper-bounded & T is upper-bounded )
proof end;

theorem :: YELLOW10:2
for S, T being non empty RelStr st [:S,T:] is lower-bounded holds
( S is lower-bounded & T is lower-bounded )
proof end;

theorem Th3: :: YELLOW10:3
for S, T being non empty antisymmetric upper-bounded RelStr holds Top [:S,T:] = [(Top S),(Top T)]
proof end;

theorem Th4: :: YELLOW10:4
for S, T being non empty antisymmetric lower-bounded RelStr holds Bottom [:S,T:] = [(),()]
proof end;

theorem Th5: :: YELLOW10:5
for S, T being non empty antisymmetric lower-bounded RelStr
for D being Subset of [:S,T:] st ( [:S,T:] is complete or ex_sup_of D,[:S,T:] ) holds
sup D = [(sup ()),(sup ())]
proof end;

theorem :: YELLOW10:6
for S, T being non empty antisymmetric upper-bounded RelStr
for D being Subset of [:S,T:] st ( [:S,T:] is complete or ex_inf_of D,[:S,T:] ) holds
inf D = [(inf ()),(inf ())]
proof end;

theorem :: YELLOW10:7
for S, T being non empty RelStr
for x, y being Element of [:S,T:] holds
( x is_<=_than {y} iff ( x 1 is_<=_than {(y 1)} & x 2 is_<=_than {(y 2)} ) )
proof end;

theorem :: YELLOW10:8
for S, T being non empty RelStr
for x, y, z being Element of [:S,T:] holds
( x is_<=_than {y,z} iff ( x 1 is_<=_than {(y 1),(z 1)} & x 2 is_<=_than {(y 2),(z 2)} ) )
proof end;

theorem :: YELLOW10:9
for S, T being non empty RelStr
for x, y being Element of [:S,T:] holds
( x is_>=_than {y} iff ( x 1 is_>=_than {(y 1)} & x 2 is_>=_than {(y 2)} ) )
proof end;

theorem :: YELLOW10:10
for S, T being non empty RelStr
for x, y, z being Element of [:S,T:] holds
( x is_>=_than {y,z} iff ( x 1 is_>=_than {(y 1),(z 1)} & x 2 is_>=_than {(y 2),(z 2)} ) )
proof end;

theorem :: YELLOW10:11
for S, T being non empty antisymmetric RelStr
for x, y being Element of [:S,T:] holds
( ex_inf_of {x,y},[:S,T:] iff ( ex_inf_of {(x 1),(y 1)},S & ex_inf_of {(x 2),(y 2)},T ) )
proof end;

theorem :: YELLOW10:12
for S, T being non empty antisymmetric RelStr
for x, y being Element of [:S,T:] holds
( ex_sup_of {x,y},[:S,T:] iff ( ex_sup_of {(x 1),(y 1)},S & ex_sup_of {(x 2),(y 2)},T ) )
proof end;

theorem Th13: :: YELLOW10:13
for S, T being antisymmetric with_infima RelStr
for x, y being Element of [:S,T:] holds
( (x "/\" y) 1 = (x 1) "/\" (y 1) & (x "/\" y) 2 = (x 2) "/\" (y 2) )
proof end;

theorem Th14: :: YELLOW10:14
for S, T being antisymmetric with_suprema RelStr
for x, y being Element of [:S,T:] holds
( (x "\/" y) 1 = (x 1) "\/" (y 1) & (x "\/" y) 2 = (x 2) "\/" (y 2) )
proof end;

theorem Th15: :: YELLOW10:15
for S, T being antisymmetric with_infima RelStr
for x1, y1 being Element of S
for x2, y2 being Element of T holds [(x1 "/\" y1),(x2 "/\" y2)] = [x1,x2] "/\" [y1,y2]
proof end;

theorem Th16: :: YELLOW10:16
for S, T being antisymmetric with_suprema RelStr
for x1, y1 being Element of S
for x2, y2 being Element of T holds [(x1 "\/" y1),(x2 "\/" y2)] = [x1,x2] "\/" [y1,y2]
proof end;

definition
let S be antisymmetric with_suprema with_infima RelStr ;
let x, y be Element of S;
:: original: is_a_complement_of
redefine pred y is_a_complement_of x;
symmetry
for y, x being Element of S st R55(S,b2,b1) holds
R55(S,b1,b2)
proof end;
end;

theorem Th17: :: YELLOW10:17
for S, T being antisymmetric bounded with_suprema with_infima RelStr
for x, y being Element of [:S,T:] holds
( x is_a_complement_of y iff ( x 1 is_a_complement_of y 1 & x 2 is_a_complement_of y 2 ) )
proof end;

theorem Th18: :: YELLOW10:18
for S, T being non empty reflexive antisymmetric up-complete RelStr
for a, c being Element of S
for b, d being Element of T st [a,b] << [c,d] holds
( a << c & b << d )
proof end;

theorem Th19: :: YELLOW10:19
for S, T being non empty up-complete Poset
for a, c being Element of S
for b, d being Element of T holds
( [a,b] << [c,d] iff ( a << c & b << d ) )
proof end;

theorem Th20: :: YELLOW10:20
for S, T being non empty reflexive antisymmetric up-complete RelStr
for x, y being Element of [:S,T:] st x << y holds
( x 1 << y 1 & x 2 << y 2 )
proof end;

theorem Th21: :: YELLOW10:21
for S, T being non empty up-complete Poset
for x, y being Element of [:S,T:] holds
( x << y iff ( x 1 << y 1 & x 2 << y 2 ) )
proof end;

theorem Th22: :: YELLOW10:22
for S, T being non empty reflexive antisymmetric up-complete RelStr
for x being Element of [:S,T:] st x is compact holds
( x 1 is compact & x 2 is compact )
proof end;

theorem Th23: :: YELLOW10:23
for S, T being non empty up-complete Poset
for x being Element of [:S,T:] st x 1 is compact & x 2 is compact holds
x is compact
proof end;

theorem Th24: :: YELLOW10:24
for S, T being antisymmetric with_infima RelStr
for X, Y being Subset of [:S,T:] holds
( proj1 (X "/\" Y) = () "/\" () & proj2 (X "/\" Y) = () "/\" () )
proof end;

theorem :: YELLOW10:25
for S, T being antisymmetric with_suprema RelStr
for X, Y being Subset of [:S,T:] holds
( proj1 (X "\/" Y) = () "\/" () & proj2 (X "\/" Y) = () "\/" () )
proof end;

theorem :: YELLOW10:26
for S, T being RelStr
for X being Subset of [:S,T:] holds downarrow X c= [:(downarrow ()),(downarrow ()):]
proof end;

theorem :: YELLOW10:27
for S, T being RelStr
for X being Subset of S
for Y being Subset of T holds [:(),():] = downarrow [:X,Y:]
proof end;

theorem Th28: :: YELLOW10:28
for S, T being RelStr
for X being Subset of [:S,T:] holds
( proj1 () c= downarrow () & proj2 () c= downarrow () )
proof end;

theorem :: YELLOW10:29
for S being RelStr
for T being reflexive RelStr
for X being Subset of [:S,T:] holds proj1 () = downarrow ()
proof end;

theorem :: YELLOW10:30
for S being reflexive RelStr
for T being RelStr
for X being Subset of [:S,T:] holds proj2 () = downarrow ()
proof end;

theorem :: YELLOW10:31
for S, T being RelStr
for X being Subset of [:S,T:] holds uparrow X c= [:(uparrow ()),(uparrow ()):]
proof end;

theorem :: YELLOW10:32
for S, T being RelStr
for X being Subset of S
for Y being Subset of T holds [:(),():] = uparrow [:X,Y:]
proof end;

theorem Th33: :: YELLOW10:33
for S, T being RelStr
for X being Subset of [:S,T:] holds
( proj1 () c= uparrow () & proj2 () c= uparrow () )
proof end;

theorem :: YELLOW10:34
for S being RelStr
for T being reflexive RelStr
for X being Subset of [:S,T:] holds proj1 () = uparrow ()
proof end;

theorem :: YELLOW10:35
for S being reflexive RelStr
for T being RelStr
for X being Subset of [:S,T:] holds proj2 () = uparrow ()
proof end;

theorem :: YELLOW10:36
for S, T being non empty RelStr
for s being Element of S
for t being Element of T holds [:(),():] = downarrow [s,t]
proof end;

theorem Th37: :: YELLOW10:37
for S, T being non empty RelStr
for x being Element of [:S,T:] holds
( proj1 () c= downarrow (x 1) & proj2 () c= downarrow (x 2) )
proof end;

theorem :: YELLOW10:38
for S being non empty RelStr
for T being non empty reflexive RelStr
for x being Element of [:S,T:] holds proj1 () = downarrow (x 1)
proof end;

theorem :: YELLOW10:39
for S being non empty reflexive RelStr
for T being non empty RelStr
for x being Element of [:S,T:] holds proj2 () = downarrow (x 2)
proof end;

theorem :: YELLOW10:40
for S, T being non empty RelStr
for s being Element of S
for t being Element of T holds [:(),():] = uparrow [s,t]
proof end;

theorem Th41: :: YELLOW10:41
for S, T being non empty RelStr
for x being Element of [:S,T:] holds
( proj1 () c= uparrow (x 1) & proj2 () c= uparrow (x 2) )
proof end;

theorem :: YELLOW10:42
for S being non empty RelStr
for T being non empty reflexive RelStr
for x being Element of [:S,T:] holds proj1 () = uparrow (x 1)
proof end;

theorem :: YELLOW10:43
for S being non empty reflexive RelStr
for T being non empty RelStr
for x being Element of [:S,T:] holds proj2 () = uparrow (x 2)
proof end;

theorem Th44: :: YELLOW10:44
for S, T being non empty up-complete Poset
for s being Element of S
for t being Element of T holds [:(),():] = waybelow [s,t]
proof end;

theorem Th45: :: YELLOW10:45
for S, T being non empty reflexive antisymmetric up-complete RelStr
for x being Element of [:S,T:] holds
( proj1 () c= waybelow (x 1) & proj2 () c= waybelow (x 2) )
proof end;

theorem Th46: :: YELLOW10:46
for S being non empty up-complete Poset
for T being non empty lower-bounded up-complete Poset
for x being Element of [:S,T:] holds proj1 () = waybelow (x 1)
proof end;

theorem Th47: :: YELLOW10:47
for S being non empty lower-bounded up-complete Poset
for T being non empty up-complete Poset
for x being Element of [:S,T:] holds proj2 () = waybelow (x 2)
proof end;

theorem :: YELLOW10:48
for S, T being non empty up-complete Poset
for s being Element of S
for t being Element of T holds [:(),():] = wayabove [s,t]
proof end;

theorem :: YELLOW10:49
for S, T being non empty reflexive antisymmetric up-complete RelStr
for x being Element of [:S,T:] holds
( proj1 () c= wayabove (x 1) & proj2 () c= wayabove (x 2) )
proof end;

theorem Th50: :: YELLOW10:50
for S, T being non empty up-complete Poset
for s being Element of S
for t being Element of T holds [:(),():] = compactbelow [s,t]
proof end;

theorem Th51: :: YELLOW10:51
for S, T being non empty reflexive antisymmetric up-complete RelStr
for x being Element of [:S,T:] holds
( proj1 () c= compactbelow (x 1) & proj2 () c= compactbelow (x 2) )
proof end;

theorem Th52: :: YELLOW10:52
for S being non empty up-complete Poset
for T being non empty lower-bounded up-complete Poset
for x being Element of [:S,T:] holds proj1 () = compactbelow (x 1)
proof end;

theorem Th53: :: YELLOW10:53
for S being non empty lower-bounded up-complete Poset
for T being non empty up-complete Poset
for x being Element of [:S,T:] holds proj2 () = compactbelow (x 2)
proof end;

registration
let S be non empty reflexive RelStr ;
cluster empty -> Open for M3( bool the carrier of S);
coherence
for b1 being Subset of S st b1 is empty holds
b1 is Open
proof end;
end;

theorem :: YELLOW10:54
for S, T being non empty reflexive antisymmetric up-complete RelStr
for X being Subset of [:S,T:] st X is Open holds
( proj1 X is Open & proj2 X is Open )
proof end;

theorem :: YELLOW10:55
for S, T being non empty up-complete Poset
for X being Subset of S
for Y being Subset of T st X is Open & Y is Open holds
[:X,Y:] is Open
proof end;

theorem :: YELLOW10:56
for S, T being non empty reflexive antisymmetric up-complete RelStr
for X being Subset of [:S,T:] st X is inaccessible holds
( proj1 X is inaccessible & proj2 X is inaccessible )
proof end;

theorem :: YELLOW10:57
for S, T being non empty reflexive antisymmetric up-complete RelStr
for X being upper Subset of S
for Y being upper Subset of T st X is inaccessible & Y is inaccessible holds
[:X,Y:] is inaccessible
proof end;

theorem :: YELLOW10:58
for S, T being non empty reflexive antisymmetric up-complete RelStr
for X being Subset of S
for Y being Subset of T st [:X,Y:] is directly_closed holds
( ( Y <> {} implies X is directly_closed ) & ( X <> {} implies Y is directly_closed ) )
proof end;

theorem :: YELLOW10:59
for S, T being non empty reflexive antisymmetric up-complete RelStr
for X being Subset of S
for Y being Subset of T st X is directly_closed & Y is directly_closed holds
[:X,Y:] is directly_closed
proof end;

theorem :: YELLOW10:60
for S, T being non empty reflexive antisymmetric up-complete RelStr
for X being Subset of [:S,T:] st X is property(S) holds
( proj1 X is property(S) & proj2 X is property(S) )
proof end;

theorem :: YELLOW10:61
for S, T being non empty up-complete Poset
for X being Subset of S
for Y being Subset of T st X is property(S) & Y is property(S) holds
[:X,Y:] is property(S)
proof end;

theorem Th62: :: YELLOW10:62
for S, T being non empty reflexive RelStr st RelStr(# the carrier of S, the InternalRel of S #) = RelStr(# the carrier of T, the InternalRel of T #) & S is /\-complete holds
T is /\-complete
proof end;

registration
let S be non empty reflexive /\-complete RelStr ;
cluster RelStr(# the carrier of S, the InternalRel of S #) -> /\-complete ;
coherence
RelStr(# the carrier of S, the InternalRel of S #) is /\-complete
by Th62;
end;

registration
let S, T be non empty reflexive /\-complete RelStr ;
cluster [:S,T:] -> /\-complete ;
coherence
[:S,T:] is /\-complete
proof end;
end;

theorem :: YELLOW10:63
for S, T being non empty reflexive RelStr st [:S,T:] is /\-complete holds
( S is /\-complete & T is /\-complete )
proof end;

registration
let S, T be non empty antisymmetric bounded complemented with_suprema with_infima RelStr ;
cluster [:S,T:] -> complemented ;
coherence
[:S,T:] is complemented
proof end;
end;

theorem :: YELLOW10:64
for S, T being antisymmetric bounded with_suprema with_infima RelStr st [:S,T:] is complemented holds
( S is complemented & T is complemented )
proof end;

registration
let S, T be non empty antisymmetric distributive with_suprema with_infima RelStr ;
cluster [:S,T:] -> distributive ;
coherence
[:S,T:] is distributive
proof end;
end;

theorem :: YELLOW10:65
for S being antisymmetric with_suprema with_infima RelStr
for T being reflexive antisymmetric with_suprema with_infima RelStr st [:S,T:] is distributive holds
S is distributive
proof end;

theorem :: YELLOW10:66
for S being reflexive antisymmetric with_suprema with_infima RelStr
for T being antisymmetric with_suprema with_infima RelStr st [:S,T:] is distributive holds
T is distributive
proof end;

registration
let S, T be meet-continuous Semilattice;
cluster [:S,T:] -> satisfying_MC ;
coherence
[:S,T:] is satisfying_MC
proof end;
end;

theorem :: YELLOW10:67
for S, T being Semilattice st [:S,T:] is meet-continuous holds
( S is meet-continuous & T is meet-continuous )
proof end;

registration
let S, T be non empty up-complete /\-complete satisfying_axiom_of_approximation Poset;
coherence
proof end;
end;

registration
let S, T be non empty /\-complete continuous Poset;
cluster [:S,T:] -> continuous ;
coherence
[:S,T:] is continuous
;
end;

theorem :: YELLOW10:68
for S, T being non empty lower-bounded up-complete Poset st [:S,T:] is continuous holds
( S is continuous & T is continuous )
proof end;

registration
let S, T be lower-bounded up-complete satisfying_axiom_K sup-Semilattice;
coherence
proof end;
end;

registration
let S, T be lower-bounded complete algebraic sup-Semilattice;
cluster [:S,T:] -> algebraic ;
coherence
[:S,T:] is algebraic
;
end;

theorem Th69: :: YELLOW10:69
for S, T being non empty lower-bounded Poset st [:S,T:] is algebraic holds
( S is algebraic & T is algebraic )
proof end;

registration
let S, T be lower-bounded arithmetic LATTICE;
cluster [:S,T:] -> arithmetic ;
coherence
[:S,T:] is arithmetic
proof end;
end;

theorem :: YELLOW10:70
for S, T being lower-bounded LATTICE st [:S,T:] is arithmetic holds
( S is arithmetic & T is arithmetic )
proof end;