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Theorem iprodclim3 25266
Description: The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that  j must not occur in  A. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
iprodclim3.1  |-  Z  =  ( ZZ>= `  M )
iprodclim3.2  |-  ( ph  ->  M  e.  ZZ )
iprodclim3.3  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq  n
(  x.  ,  ( k  e.  Z  |->  A ) )  ~~>  y ) )
iprodclim3.4  |-  ( ph  ->  F  e.  dom  ~~>  )
iprodclim3.5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
iprodclim3.6  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  prod_ k  e.  ( M ... j ) A )
Assertion
Ref Expression
iprodclim3  |-  ( ph  ->  F  ~~>  prod_ k  e.  Z A )
Distinct variable groups:    A, j    A, n, y    j, F   
j, k, ph    k, n,
ph, y    j, M    y, M    ph, n, y    j, Z, k    n, Z, y   
k, M
Allowed substitution hints:    A( k)    F( y, k, n)    M( n)

Proof of Theorem iprodclim3
Dummy variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iprodclim3.4 . . 3  |-  ( ph  ->  F  e.  dom  ~~>  )
2 climdm 12303 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 189 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 prodfc 25224 . . . 4  |-  prod_ m  e.  Z ( ( k  e.  Z  |->  A ) `
 m )  = 
prod_ k  e.  Z A
5 iprodclim3.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
6 iprodclim3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
7 iprodclim3.3 . . . . 5  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq  n
(  x.  ,  ( k  e.  Z  |->  A ) )  ~~>  y ) )
8 eqidd 2405 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
9 iprodclim3.5 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
10 eqid 2404 . . . . . . 7  |-  ( k  e.  Z  |->  A )  =  ( k  e.  Z  |->  A )
119, 10fmptd 5852 . . . . . 6  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
1211ffvelrnda 5829 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
135, 6, 7, 8, 12iprod 25217 . . . 4  |-  ( ph  ->  prod_ m  e.  Z
( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq  M (  x.  ,  ( k  e.  Z  |->  A ) ) ) )
144, 13syl5eqr 2450 . . 3  |-  ( ph  ->  prod_ k  e.  Z A  =  (  ~~>  `  seq  M (  x.  ,  ( k  e.  Z  |->  A ) ) ) )
15 seqex 11280 . . . . . . 7  |-  seq  M
(  x.  ,  ( k  e.  Z  |->  A ) )  e.  _V
1615a1i 11 . . . . . 6  |-  ( ph  ->  seq  M (  x.  ,  ( k  e.  Z  |->  A ) )  e.  _V )
17 iprodclim3.6 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  prod_ k  e.  ( M ... j ) A )
18 fzssuz 11049 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1918, 5sseqtr4i 3341 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
20 resmpt 5150 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2119, 20ax-mp 8 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2221fveq1i 5688 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
23 fvres 5704 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2422, 23syl5reqr 2451 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524prodeq2i 25198 . . . . . . . . 9  |-  prod_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
prod_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 prodfc 25224 . . . . . . . . 9  |-  prod_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
prod_ k  e.  ( M ... j ) A
2725, 26eqtri 2424 . . . . . . . 8  |-  prod_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
prod_ k  e.  ( M ... j ) A
28 eqidd 2405 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
29 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3029, 5syl6eleq 2494 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
31 elfzuz 11011 . . . . . . . . . . . 12  |-  ( m  e.  ( M ... j )  ->  m  e.  ( ZZ>= `  M )
)
3231, 5syl6eleqr 2495 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  m  e.  Z )
3332, 12sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( M ... j ) )  ->  ( (
k  e.  Z  |->  A ) `  m )  e.  CC )
3433adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
3528, 30, 34fprodser 25228 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  prod_ m  e.  ( M ... j ) ( ( k  e.  Z  |->  A ) `  m )  =  (  seq  M
(  x.  ,  ( k  e.  Z  |->  A ) ) `  j
) )
3627, 35syl5eqr 2450 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  prod_ k  e.  ( M ... j ) A  =  (  seq  M (  x.  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3717, 36eqtr2d 2437 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  x.  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
385, 16, 1, 6, 37climeq 12316 . . . . 5  |-  ( ph  ->  (  seq  M (  x.  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
3938iotabidv 5398 . . . 4  |-  ( ph  ->  ( iota x  seq  M (  x.  ,  ( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
40 df-fv 5421 . . . 4  |-  (  ~~>  `  seq  M (  x.  ,  ( k  e.  Z  |->  A ) ) )  =  ( iota x  seq  M (  x.  ,  ( k  e.  Z  |->  A ) )  ~~>  x )
41 df-fv 5421 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4239, 40, 413eqtr4g 2461 . . 3  |-  ( ph  ->  (  ~~>  `  seq  M (  x.  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4314, 42eqtrd 2436 . 2  |-  ( ph  ->  prod_ k  e.  Z A  =  (  ~~>  `  F
) )
443, 43breqtrrd 4198 1  |-  ( ph  ->  F  ~~>  prod_ k  e.  Z A )
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    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   dom cdm 4837    |` cres 4839   iotacio 5375   ` 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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