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Theorem iprodmul 25269
Description: Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
iprodmul.1  |-  Z  =  ( ZZ>= `  M )
iprodmul.2  |-  ( ph  ->  M  e.  ZZ )
iprodmul.3  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq  n
(  x.  ,  F
)  ~~>  y ) )
iprodmul.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
iprodmul.5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
iprodmul.6  |-  ( ph  ->  E. m  e.  Z  E. z ( z  =/=  0  /\  seq  m
(  x.  ,  G
)  ~~>  z ) )
iprodmul.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )
iprodmul.8  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
Assertion
Ref Expression
iprodmul  |-  ( ph  ->  prod_ k  e.  Z
( A  x.  B
)  =  ( prod_
k  e.  Z A  x.  prod_ k  e.  Z B ) )
Distinct variable groups:    A, n, y    B, m, z    k, F, m, n, y, z   
k, G, m, n, y, z    ph, k,
y, z    k, M, m, n    ph, m, y   
y, M    z, m, M    ph, n, y, z   
k, Z, m, n, y, z
Allowed substitution hints:    A( z, k, m)    B( y, k, n)

Proof of Theorem iprodmul
Dummy variables  j 
a  p  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iprodmul.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 iprodmul.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 iprodmul.3 . . . 4  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq  n
(  x.  ,  F
)  ~~>  y ) )
4 iprodmul.4 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
5 iprodmul.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
64, 5eqeltrd 2478 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
7 iprodmul.6 . . . 4  |-  ( ph  ->  E. m  e.  Z  E. z ( z  =/=  0  /\  seq  m
(  x.  ,  G
)  ~~>  z ) )
8 iprodmul.7 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )
9 iprodmul.8 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
108, 9eqeltrd 2478 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
11 fveq2 5687 . . . . . . 7  |-  ( a  =  k  ->  ( F `  a )  =  ( F `  k ) )
12 fveq2 5687 . . . . . . 7  |-  ( a  =  k  ->  ( G `  a )  =  ( G `  k ) )
1311, 12oveq12d 6058 . . . . . 6  |-  ( a  =  k  ->  (
( F `  a
)  x.  ( G `
 a ) )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
14 eqid 2404 . . . . . 6  |-  ( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a ) ) )  =  ( a  e.  Z  |->  ( ( F `
 a )  x.  ( G `  a
) ) )
15 ovex 6065 . . . . . 6  |-  ( ( F `  k )  x.  ( G `  k ) )  e. 
_V
1613, 14, 15fvmpt 5765 . . . . 5  |-  ( k  e.  Z  ->  (
( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a )
) ) `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )
1716adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a )
) ) `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )
181, 3, 6, 7, 10, 17ntrivcvgmul 25183 . . 3  |-  ( ph  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq  p
(  x.  ,  ( a  e.  Z  |->  ( ( F `  a
)  x.  ( G `
 a ) ) ) )  ~~>  w ) )
19 fveq2 5687 . . . . . . . . . 10  |-  ( m  =  a  ->  ( F `  m )  =  ( F `  a ) )
20 fveq2 5687 . . . . . . . . . 10  |-  ( m  =  a  ->  ( G `  m )  =  ( G `  a ) )
2119, 20oveq12d 6058 . . . . . . . . 9  |-  ( m  =  a  ->  (
( F `  m
)  x.  ( G `
 m ) )  =  ( ( F `
 a )  x.  ( G `  a
) ) )
2221cbvmptv 4260 . . . . . . . 8  |-  ( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m ) ) )  =  ( a  e.  Z  |->  ( ( F `
 a )  x.  ( G `  a
) ) )
23 seqeq3 11283 . . . . . . . 8  |-  ( ( m  e.  Z  |->  ( ( F `  m
)  x.  ( G `
 m ) ) )  =  ( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a ) ) )  ->  seq  p (  x.  ,  ( m  e.  Z  |->  ( ( F `
 m )  x.  ( G `  m
) ) ) )  =  seq  p (  x.  ,  ( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a ) ) ) ) )
2422, 23ax-mp 8 . . . . . . 7  |-  seq  p
(  x.  ,  ( m  e.  Z  |->  ( ( F `  m
)  x.  ( G `
 m ) ) ) )  =  seq  p (  x.  , 
( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a )
) ) )
2524breq1i 4179 . . . . . 6  |-  (  seq  p (  x.  , 
( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m )
) ) )  ~~>  w  <->  seq  p (  x.  ,  ( a  e.  Z  |->  ( ( F `  a )  x.  ( G `  a ) ) ) )  ~~>  w )
2625anbi2i 676 . . . . 5  |-  ( ( w  =/=  0  /\ 
seq  p (  x.  ,  ( m  e.  Z  |->  ( ( F `
 m )  x.  ( G `  m
) ) ) )  ~~>  w )  <->  ( w  =/=  0  /\  seq  p
(  x.  ,  ( a  e.  Z  |->  ( ( F `  a
)  x.  ( G `
 a ) ) ) )  ~~>  w ) )
2726exbii 1589 . . . 4  |-  ( E. w ( w  =/=  0  /\  seq  p
(  x.  ,  ( m  e.  Z  |->  ( ( F `  m
)  x.  ( G `
 m ) ) ) )  ~~>  w )  <->  E. w ( w  =/=  0  /\  seq  p
(  x.  ,  ( a  e.  Z  |->  ( ( F `  a
)  x.  ( G `
 a ) ) ) )  ~~>  w ) )
2827rexbii 2691 . . 3  |-  ( E. p  e.  Z  E. w ( w  =/=  0  /\  seq  p
(  x.  ,  ( m  e.  Z  |->  ( ( F `  m
)  x.  ( G `
 m ) ) ) )  ~~>  w )  <->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq  p
(  x.  ,  ( a  e.  Z  |->  ( ( F `  a
)  x.  ( G `
 a ) ) ) )  ~~>  w ) )
2918, 28sylibr 204 . 2  |-  ( ph  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq  p
(  x.  ,  ( m  e.  Z  |->  ( ( F `  m
)  x.  ( G `
 m ) ) ) )  ~~>  w ) )
30 simpr 448 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
316, 10mulcld 9064 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( G `
 k ) )  e.  CC )
32 fveq2 5687 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
33 fveq2 5687 . . . . . 6  |-  ( m  =  k  ->  ( G `  m )  =  ( G `  k ) )
3432, 33oveq12d 6058 . . . . 5  |-  ( m  =  k  ->  (
( F `  m
)  x.  ( G `
 m ) )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
35 eqid 2404 . . . . 5  |-  ( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m ) ) )  =  ( m  e.  Z  |->  ( ( F `
 m )  x.  ( G `  m
) ) )
3634, 35fvmptg 5763 . . . 4  |-  ( ( k  e.  Z  /\  ( ( F `  k )  x.  ( G `  k )
)  e.  CC )  ->  ( ( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m ) ) ) `
 k )  =  ( ( F `  k )  x.  ( G `  k )
) )
3730, 31, 36syl2anc 643 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m )
) ) `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )
384, 8oveq12d 6058 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( A  x.  B ) )
3937, 38eqtrd 2436 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m )
) ) `  k
)  =  ( A  x.  B ) )
405, 9mulcld 9064 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  x.  B )  e.  CC )
411, 2, 3, 4, 5iprodclim2 25265 . . 3  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  prod_ k  e.  Z A )
42 seqex 11280 . . . 4  |-  seq  M
(  x.  ,  ( m  e.  Z  |->  ( ( F `  m
)  x.  ( G `
 m ) ) ) )  e.  _V
4342a1i 11 . . 3  |-  ( ph  ->  seq  M (  x.  ,  ( m  e.  Z  |->  ( ( F `
 m )  x.  ( G `  m
) ) ) )  e.  _V )
441, 2, 7, 8, 9iprodclim2 25265 . . 3  |-  ( ph  ->  seq  M (  x.  ,  G )  ~~>  prod_ k  e.  Z B )
451, 2, 6prodf 25168 . . . 4  |-  ( ph  ->  seq  M (  x.  ,  F ) : Z --> CC )
4645ffvelrnda 5829 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  x.  ,  F ) `  j
)  e.  CC )
471, 2, 10prodf 25168 . . . 4  |-  ( ph  ->  seq  M (  x.  ,  G ) : Z --> CC )
4847ffvelrnda 5829 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  x.  ,  G ) `  j
)  e.  CC )
49 simpr 448 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
5049, 1syl6eleq 2494 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
51 elfzuz 11011 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
5251, 1syl6eleqr 2495 . . . . . 6  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
5352, 6sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
5453adantlr 696 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ( F `  k )  e.  CC )
5552, 10sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... j ) )  ->  ( G `  k )  e.  CC )
5655adantlr 696 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ( G `  k )  e.  CC )
5737adantlr 696 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m )
) ) `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )
5852, 57sylan2 461 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  (
( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m )
) ) `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )
5950, 54, 56, 58prodfmul 25171 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  x.  , 
( m  e.  Z  |->  ( ( F `  m )  x.  ( G `  m )
) ) ) `  j )  =  ( (  seq  M (  x.  ,  F ) `
 j )  x.  (  seq  M (  x.  ,  G ) `
 j ) ) )
601, 2, 41, 43, 44, 46, 48, 59climmul 12381 . 2  |-  ( ph  ->  seq  M (  x.  ,  ( m  e.  Z  |->  ( ( F `
 m )  x.  ( G `  m
) ) ) )  ~~>  ( prod_ k  e.  Z A  x.  prod_ k  e.  Z B ) )
611, 2, 29, 39, 40, 60iprodclim 25264 1  |-  ( ph  ->  prod_ k  e.  Z
( A  x.  B
)  =  ( prod_
k  e.  Z A  x.  prod_ k  e.  Z B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    x. cmul 8951   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278    ~~> cli 12233   prod_cprod 25184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-prod 25185
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