:: Complex Sequences
:: by Agnieszka Banachowicz and Anna Winnicka
::
:: Received November 5, 1993
:: Copyright (c) 1993 Association of Mizar Users

definition

mode Complex_Sequence is sequence of COMPLEX ;

end;

theorem Th1: :: COMSEQ_1:1

for f being Function holds
( f is Complex_Sequence iff ( dom f = NAT & ( for x being set st x in NAT holds
f . x is Element of COMPLEX ) ) )

proof end;

theorem Th2: :: COMSEQ_1:2

for f being Function holds
( f is Complex_Sequence iff ( dom f = NAT & ( for n being Element of NAT holds f . n is Element of COMPLEX ) ) )

proof end;

scheme :: COMSEQ_1:sch 1
ExComplexSeq{ F₁( set ) -> complex number } :

ex seq being Complex_Sequence st
for n being Element of NAT holds seq . n = F₁(n)

proof end;

notation

let f be complex-valued Relation;
synonym non-zero f for non-empty ;

end;

registration

cluster non-zero Element of K21(K22(NAT ,COMPLEX ));
existence
ex b₁ being Complex_Sequence st b₁ is non-zero

proof end;

end;

theorem Th3: :: COMSEQ_1:3

for seq being Complex_Sequence holds
( seq is non-zero iff for x being set st x in NAT holds
seq . x <> 0c )

proof end;

theorem Th4: :: COMSEQ_1:4

for seq being Complex_Sequence holds
( seq is non-zero iff for n being Element of NAT holds seq . n <> 0c )

proof end;

theorem :: COMSEQ_1:5

for IT being non-zero Complex_Sequence holds rng IT c= COMPLEX \ {0c }

proof end;

theorem :: COMSEQ_1:6

canceled;

theorem :: COMSEQ_1:7

for r being Element of COMPLEX ex seq being Complex_Sequence st rng seq = {r}

proof end;

theorem :: COMSEQ_1:8

canceled;

theorem Th9: :: COMSEQ_1:9

for seq1, seq2, seq3 being Complex_Sequence holds (seq1 + seq2) + seq3 = seq1 + (seq2 + seq3)

proof end;

theorem :: COMSEQ_1:10

canceled;

theorem Th11: :: COMSEQ_1:11

for seq1, seq2, seq3 being Complex_Sequence holds (seq1 (#) seq2) (#) seq3 = seq1 (#) (seq2 (#) seq3)

proof end;

theorem Th12: :: COMSEQ_1:12

for seq1, seq2, seq3 being Complex_Sequence holds (seq1 + seq2) (#) seq3 = (seq1 (#) seq3) + (seq2 (#) seq3)

proof end;

theorem :: COMSEQ_1:13

for seq3, seq1, seq2 being Complex_Sequence holds seq3 (#) (seq1 + seq2) = (seq3 (#) seq1) + (seq3 (#) seq2) by Th12;

theorem :: COMSEQ_1:14

for seq being Complex_Sequence holds - seq = (- 1r ) (#) seq ;

theorem Th15: :: COMSEQ_1:15

for r being Element of COMPLEX
for seq1, seq2 being Complex_Sequence holds r (#) (seq1 (#) seq2) = (r (#) seq1) (#) seq2

proof end;

theorem Th16: :: COMSEQ_1:16

for r being Element of COMPLEX
for seq1, seq2 being Complex_Sequence holds r (#) (seq1 (#) seq2) = seq1 (#) (r (#) seq2)

proof end;

theorem Th17: :: COMSEQ_1:17

for seq1, seq2, seq3 being Complex_Sequence holds (seq1 - seq2) (#) seq3 = (seq1 (#) seq3) - (seq2 (#) seq3)

proof end;

theorem :: COMSEQ_1:18

for seq3, seq1, seq2 being Complex_Sequence holds (seq3 (#) seq1) - (seq3 (#) seq2) = seq3 (#) (seq1 - seq2) by Th17;

theorem Th19: :: COMSEQ_1:19

for r being Element of COMPLEX
for seq1, seq2 being Complex_Sequence holds r (#) (seq1 + seq2) = (r (#) seq1) + (r (#) seq2)

proof end;

theorem Th20: :: COMSEQ_1:20

for r, p being Element of COMPLEX
for seq being Complex_Sequence holds (r * p) (#) seq = r (#) (p (#) seq)

proof end;

theorem Th21: :: COMSEQ_1:21

for r being Element of COMPLEX
for seq1, seq2 being Complex_Sequence holds r (#) (seq1 - seq2) = (r (#) seq1) - (r (#) seq2)

proof end;

theorem :: COMSEQ_1:22

for r being Element of COMPLEX
for seq1, seq being Complex_Sequence holds r (#) (seq1 /" seq) = (r (#) seq1) /" seq

proof end;

theorem :: COMSEQ_1:23

for seq1, seq2, seq3 being Complex_Sequence holds seq1 - (seq2 + seq3) = (seq1 - seq2) - seq3

proof end;

theorem :: COMSEQ_1:24

for seq being Complex_Sequence holds 1r (#) seq = seq

proof end;

theorem :: COMSEQ_1:25

for seq being Complex_Sequence holds - (- seq) = seq ;

theorem :: COMSEQ_1:26

for seq1, seq2 being Complex_Sequence holds seq1 - (- seq2) = seq1 + seq2 ;

theorem :: COMSEQ_1:27

for seq1, seq2, seq3 being Complex_Sequence holds seq1 - (seq2 - seq3) = (seq1 - seq2) + seq3

proof end;

theorem :: COMSEQ_1:28

for seq1, seq2, seq3 being Complex_Sequence holds seq1 + (seq2 - seq3) = (seq1 + seq2) - seq3

proof end;

theorem :: COMSEQ_1:29

for seq1, seq2 being Complex_Sequence holds
( (- seq1) (#) seq2 = - (seq1 (#) seq2) & seq1 (#) (- seq2) = - (seq1 (#) seq2) )

proof end;

theorem Th30: :: COMSEQ_1:30

for seq being Complex_Sequence st seq is non-zero holds
seq " is non-zero

proof end;

theorem :: COMSEQ_1:31

for seq being Complex_Sequence holds (seq " ) " = seq ;

theorem Th32: :: COMSEQ_1:32

for seq, seq1 being Complex_Sequence holds
( ( seq is non-zero & seq1 is non-zero ) iff seq (#) seq1 is non-zero )

proof end;

theorem Th33: :: COMSEQ_1:33

for seq, seq1 being Complex_Sequence holds (seq " ) (#) (seq1 " ) = (seq (#) seq1) "

proof end;

theorem :: COMSEQ_1:34

for seq, seq1 being Complex_Sequence st seq is non-zero holds
(seq1 /" seq) (#) seq = seq1

proof end;

theorem :: COMSEQ_1:35

for seq', seq, seq1', seq1 being Complex_Sequence holds (seq' /" seq) (#) (seq1' /" seq1) = (seq' (#) seq1') /" (seq (#) seq1)

proof end;

theorem :: COMSEQ_1:36

for seq, seq1 being Complex_Sequence st seq is non-zero & seq1 is non-zero holds
seq /" seq1 is non-zero

proof end;

theorem Th37: :: COMSEQ_1:37

for seq, seq1 being Complex_Sequence holds (seq /" seq1) " = seq1 /" seq

proof end;

theorem :: COMSEQ_1:38

for seq2, seq1, seq being Complex_Sequence holds seq2 (#) (seq1 /" seq) = (seq2 (#) seq1) /" seq

proof end;

theorem :: COMSEQ_1:39

for seq2, seq, seq1 being Complex_Sequence holds seq2 /" (seq /" seq1) = (seq2 (#) seq1) /" seq

proof end;

theorem Th40: :: COMSEQ_1:40

for seq1, seq2, seq being Complex_Sequence st seq1 is non-zero holds
seq2 /" seq = (seq2 (#) seq1) /" (seq (#) seq1)

proof end;

theorem Th41: :: COMSEQ_1:41

for r being Element of COMPLEX
for seq being Complex_Sequence st r <> 0c & seq is non-zero holds
r (#) seq is non-zero

proof end;

theorem :: COMSEQ_1:42

for seq being Complex_Sequence st seq is non-zero holds
- seq is non-zero

proof end;

theorem Th43: :: COMSEQ_1:43

for r being Element of COMPLEX
for seq being Complex_Sequence holds (r (#) seq) " = (r " ) (#) (seq " )

proof end;

theorem Th44: :: COMSEQ_1:44

for seq being Complex_Sequence st seq is non-zero holds
(- seq) " = (- 1r ) (#) (seq " )

proof end;

theorem :: COMSEQ_1:45

for seq, seq1 being Complex_Sequence st seq is non-zero holds
( - (seq1 /" seq) = (- seq1) /" seq & seq1 /" (- seq) = - (seq1 /" seq) )

proof end;

theorem :: COMSEQ_1:46

for seq1, seq, seq1' being Complex_Sequence holds
( (seq1 /" seq) + (seq1' /" seq) = (seq1 + seq1') /" seq & (seq1 /" seq) - (seq1' /" seq) = (seq1 - seq1') /" seq )

proof end;

theorem :: COMSEQ_1:47

for seq, seq', seq1, seq1' being Complex_Sequence st seq is non-zero & seq' is non-zero holds
( (seq1 /" seq) + (seq1' /" seq') = ((seq1 (#) seq') + (seq1' (#) seq)) /" (seq (#) seq') & (seq1 /" seq) - (seq1' /" seq') = ((seq1 (#) seq') - (seq1' (#) seq)) /" (seq (#) seq') )

proof end;

theorem :: COMSEQ_1:48

for seq1', seq, seq', seq1 being Complex_Sequence holds (seq1' /" seq) /" (seq' /" seq1) = (seq1' (#) seq1) /" (seq (#) seq')

proof end;

theorem Th49: :: COMSEQ_1:49

for seq, seq' being Complex_Sequence holds |.(seq (#) seq').| = |.seq.| (#) |.seq'.|

proof end;

theorem :: COMSEQ_1:50

for seq being Complex_Sequence st seq is non-zero holds
|.seq.| is non-zero

proof end;

theorem Th51: :: COMSEQ_1:51

for seq being Complex_Sequence holds |.seq.| " = |.(seq " ).|

proof end;

theorem :: COMSEQ_1:52

for seq', seq being Complex_Sequence holds |.(seq' /" seq).| = |.seq'.| /" |.seq.|

proof end;

theorem :: COMSEQ_1:53

for r being Element of COMPLEX
for seq being Complex_Sequence holds |.(r (#) seq).| = |.r.| (#) |.seq.|

proof end;