:: Complex Sequences
:: by Agnieszka Banachowicz and Anna Winnicka
::
:: Received November 5, 1993
:: Copyright (c) 1993-2011 Association of Mizar Users

begin

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 Relation-like non-zero NAT -defined COMPLEX -valued Function-like non zero V14( NAT ) V15( NAT , COMPLEX ) complex-valued 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 seq9, seq, seq19, seq1 being Complex_Sequence holds (seq9 /" seq) (#) (seq19 /" seq1) = (seq9 (#) seq19) /" (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, seq19 being Complex_Sequence holds
( (seq1 /" seq) + (seq19 /" seq) = (seq1 + seq19) /" seq & (seq1 /" seq) - (seq19 /" seq) = (seq1 - seq19) /" seq )

proof end;

theorem :: COMSEQ_1:47

for seq, seq9, seq1, seq19 being Complex_Sequence st seq is non-zero & seq9 is non-zero holds
( (seq1 /" seq) + (seq19 /" seq9) = ((seq1 (#) seq9) + (seq19 (#) seq)) /" (seq (#) seq9) & (seq1 /" seq) - (seq19 /" seq9) = ((seq1 (#) seq9) - (seq19 (#) seq)) /" (seq (#) seq9) )

proof end;

theorem :: COMSEQ_1:48

for seq19, seq, seq9, seq1 being Complex_Sequence holds (seq19 /" seq) /" (seq9 /" seq1) = (seq19 (#) seq1) /" (seq (#) seq9)

proof end;

theorem Th49: :: COMSEQ_1:49

for seq, seq9 being Complex_Sequence holds |.(seq (#) seq9).| = |.seq.| (#) |.seq9.|

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 seq9, seq being Complex_Sequence holds |.(seq9 /" seq).| = |.seq9.| /" |.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;