:: Some Topological Properties of Cells in $R^2$
:: by Yatsuka Nakamura and Andrzej Trybulec
::
:: Received July 22, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, REAL_1, XBOOLE_0, PRE_TOPC, SQUARE_1,
XXREAL_0, CARD_1, RELAT_2, TARSKI, CONNSP_1, SETFAM_1, RCOMP_1, CONNSP_3,
ZFMISC_1, STRUCT_0, FUNCT_1, GOBOARD5, RELAT_1, MATRIX_1, EUCLID,
TOPREAL1, RLTOPSP1, ARYTM_3, GOBOARD2, TREES_1, ARYTM_1, FINSEQ_1,
MCART_1, GOBOARD1, TOPS_1, PCOMPS_1, METRIC_1, NAT_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SEQM_3, TOPREAL1, GOBOARD1,
ORDINAL1, XXREAL_0, NUMBERS, BINOP_1, XCMPLX_0, XREAL_0, REAL_1, NAT_1,
NAT_D, SQUARE_1, MATRIX_0, STRUCT_0, PRE_TOPC, FINSEQ_1, MATRIX_1,
METRIC_1, TOPS_1, CONNSP_1, PCOMPS_1, RLVECT_1, RLTOPSP1, EUCLID,
GOBOARD2, GOBOARD5, CONNSP_3;
constructors SETFAM_1, REAL_1, SQUARE_1, TOPS_1, CONNSP_1, PCOMPS_1, GOBOARD2,
GOBOARD5, CONNSP_3, GOBOARD1, NAT_D, FUNCSDOM, CONVEX1;
registrations SUBSET_1, RELSET_1, XXREAL_0, XREAL_0, STRUCT_0, PRE_TOPC,
EUCLID, GOBOARD1, GOBOARD2, GOBOARD5, RLTOPSP1, SQUARE_1, ORDINAL1,
NAT_1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
begin
reserve i,j,k for Nat,
r,s,r1,r2,s1,s2,sb,tb for Real,
x for set,
GX for non empty TopSpace;
theorem :: GOBRD11:1
for A being Subset of GX, p being Point of GX st p in A & A is
connected holds A c= Component_of p;
theorem :: GOBRD11:2
for A,B,C being Subset of GX st C is a_component & A c= C & B is
connected & Cl A meets Cl B holds B c= C;
reserve GZ for non empty TopSpace;
theorem :: GOBRD11:3
for A,B being Subset of GZ st A is a_component & B
is a_component holds A \/ B is a_union_of_components of GZ;
theorem :: GOBRD11:4
for B1,B2,V being Subset of GX holds Down(B1 \/ B2,V)=Down(B1,V) \/
Down(B2,V);
theorem :: GOBRD11:5
for B1,B2,V being Subset of GX holds Down(B1 /\ B2,V)=Down(B1,V) /\
Down(B2,V);
reserve f for non constant standard special_circular_sequence,
G for non empty-yielding Matrix of TOP-REAL 2;
theorem :: GOBRD11:6
(L~f)` <> {};
registration
let f;
cluster (L~f)` -> non empty;
end;
theorem :: GOBRD11:7
for f holds the carrier of TOP-REAL 2 = union {cell(GoB f,i,j):i<=len
GoB f & j<=width GoB f};
theorem :: GOBRD11:8
for P1,P2 being Subset of TOP-REAL 2 st P1={ |[r,s]| : s <= s1 }
& P2={ |[r2,s2]| : s2 > s1 } holds P1=P2`;
theorem :: GOBRD11:9
for P1,P2 being Subset of TOP-REAL 2 st P1={ |[r,s]| : s >= s1 }
& P2={ |[r2,s2]| : s2 < s1 } holds P1=P2`;
theorem :: GOBRD11:10
for P1,P2 being Subset of TOP-REAL 2 st
P1={ |[s,r]| where s,r is Real: s >= s1 }
& P2={ |[s2,r2]| where s2,r2 is Real: s2 < s1 } holds P1=P2`;
theorem :: GOBRD11:11
for P1,P2 being Subset of TOP-REAL 2 st
P1={ |[s,r]| where s,r is Real: s <= s1 }
& P2={ |[s2,r2]| where s2,r2 is Real: s2 > s1 } holds P1=P2`;
theorem :: GOBRD11:12
for P being Subset of TOP-REAL 2, s1 st P= { |[r,s]| : s <= s1 }
holds P is closed;
theorem :: GOBRD11:13
for P being Subset of TOP-REAL 2,s1 st P={ |[r,s]| : s1 <= s }
holds P is closed;
theorem :: GOBRD11:14
for P being Subset of TOP-REAL 2, s1 st
P= { |[s,r]| where s,r is Real: s <= s1 }
holds P is closed;
theorem :: GOBRD11:15
for P being Subset of TOP-REAL 2,s1 st
P={ |[s,r]| where s,r is Real: s1 <= s }
holds P is closed;
theorem :: GOBRD11:16
for G being Matrix of TOP-REAL 2 holds h_strip(G,j) is closed;
theorem :: GOBRD11:17
for G being Matrix of TOP-REAL 2 holds v_strip(G,j) is closed;
theorem :: GOBRD11:18
G is X_equal-in-line implies v_strip(G,0) = { |[r,s]| : r <= G*( 1,1)`1 };
theorem :: GOBRD11:19
G is X_equal-in-line implies v_strip(G,len G) = { |[r,s]| : G*(
len G,1)`1 <= r };
theorem :: GOBRD11:20
G is X_equal-in-line & 1 <= i & i < len G implies v_strip(G,i) =
{ |[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1 };
theorem :: GOBRD11:21
G is Y_equal-in-column implies h_strip(G,0) = { |[r,s]| : s <= G *(1,1)`2 };
theorem :: GOBRD11:22
G is Y_equal-in-column implies h_strip(G,width G) = { |[r,s]| :
G*(1,width G)`2 <= s };
theorem :: GOBRD11:23
G is Y_equal-in-column & 1 <= j & j < width G implies h_strip(G,
j) = { |[r,s]| : G*(1,j)`2 <= s & s <= G*(1,j+1)`2 };
reserve G for non empty-yielding X_equal-in-line Y_equal-in-column Matrix of
TOP-REAL 2;
theorem :: GOBRD11:24
cell(G,0,0) = { |[r,s]| : r <= G*(1,1)`1 & s <= G*(1,1)`2 };
theorem :: GOBRD11:25
cell(G,0,width G) = { |[r,s]| : r <= G*(1,1)`1 & G*(1,width G)`2 <= s };
theorem :: GOBRD11:26
1 <= j & j < width G implies cell(G,0,j) = { |[r,s]| : r <= G*(1
,1)`1 & G*(1,j)`2 <= s & s <= G*(1,j+1)`2 };
theorem :: GOBRD11:27
cell(G,len G,0) = { |[r,s]| : G*(len G,1)`1 <= r & s <= G*(1,1) `2 };
theorem :: GOBRD11:28
cell(G,len G,width G) = { |[r,s]| : G*(len G,1)`1 <= r & G*(1,
width G)`2 <= s };
theorem :: GOBRD11:29
1 <= j & j < width G implies cell(G,len G,j) = { |[r,s]| : G*(
len G,1)`1 <= r & G*(1,j)`2 <= s & s <= G*(1,j+1)`2 };
theorem :: GOBRD11:30
1 <= i & i < len G implies cell(G,i,0) = { |[r,s]|: G*(i,1)`1 <=
r & r <= G*(i+1,1)`1 & s <= G* (1,1)`2 };
theorem :: GOBRD11:31
1 <= i & i < len G implies cell(G,i,width G) = { |[r,s]|: G*(i,1
)`1 <= r & r <= G*(i+1,1)`1 & G*(1,width G)`2 <= s };
theorem :: GOBRD11:32
1 <= i & i < len G & 1 <= j & j < width G implies cell(G,i,j) =
{ |[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,j)`2 <= s & s <= G*(1,j+1)
`2 };
theorem :: GOBRD11:33
for G being Matrix of TOP-REAL 2 holds cell(G,i,j) is closed;
theorem :: GOBRD11:34
for G being non empty-yielding Matrix of TOP-REAL 2 holds 1<=len
G & 1<=width G;
theorem :: GOBRD11:35
for G being Go-board holds i<=len G & j<=width G implies cell(G,i,j)=
Cl Int cell(G,i,j);