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Theorem fprodcvg 25209
Description: The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
prodmo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
prodrb.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fprodcvg.4  |-  ( ph  ->  A  C_  ( M ... N ) )
Assertion
Ref Expression
fprodcvg  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  (  seq 
M (  x.  ,  F ) `  N
) )
Distinct variable groups:    A, k    k, F    ph, k    k, M   
k, N
Allowed substitution hint:    B( k)

Proof of Theorem fprodcvg
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 prodrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 10452 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 16 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 11280 . . 3  |-  seq  M
(  x.  ,  F
)  e.  _V
65a1i 11 . 2  |-  ( ph  ->  seq  M (  x.  ,  F )  e. 
_V )
7 eqid 2404 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 10449 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 10452 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1110adantl 453 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ZZ )
12 iftrue 3705 . . . . . . . . . 10  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
1312adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  =  B )
14 prodmo.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
1514adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
1613, 15eqeltrd 2478 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1716ex 424 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC ) )
18 iffalse 3706 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
19 ax-1cn 9004 . . . . . . . 8  |-  1  e.  CC
2018, 19syl6eqel 2492 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
2117, 20pm2.61d1 153 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  1 )  e.  CC )
22 prodmo.1 . . . . . . 7  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
2322fvmpt2 5771 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
2411, 21, 23syl2anc 643 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  1 ) )
2524, 21eqeltrd 2478 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
267, 9, 25prodf 25168 . . 3  |-  ( ph  ->  seq  M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
2726, 2ffvelrnd 5830 . 2  |-  ( ph  ->  (  seq  M (  x.  ,  F ) `
 N )  e.  CC )
28 mulid1 9044 . . . . 5  |-  ( m  e.  CC  ->  (
m  x.  1 )  =  m )
2928adantl 453 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  CC )  ->  ( m  x.  1 )  =  m )
302adantr 452 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
31 simpr 448 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
329adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
3325adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
347, 32, 33prodf 25168 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  seq  M (  x.  ,  F ) : ( ZZ>= `  M
) --> CC )
3534, 30ffvelrnd 5830 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  x.  ,  F
) `  N )  e.  CC )
36 elfzuz 11011 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
37 eluzelz 10452 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  ( N  +  1 ) )  ->  m  e.  ZZ )
3837adantl 453 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ZZ )
39 fprodcvg.4 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  ( M ... N ) )
4039sseld 3307 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
41 fznuz 11084 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
4240, 41syl6 31 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
4342con2d 109 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4443imp 419 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4538, 44eldifd 3291 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
46 fveq2 5687 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4746eqeq1d 2412 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  1  <->  ( F `  m )  =  1 ) )
48 eldifi 3429 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
49 eldifn 3430 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
5049, 18syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
5150, 19syl6eqel 2492 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
5248, 51, 23syl2anc 643 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
5352, 50eqtrd 2436 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
5447, 53vtoclga 2977 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  1 )
5545, 54syl 16 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  1 )
5636, 55sylan2 461 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  1 )
5756adantlr 696 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  1 )
5829, 30, 31, 35, 57seqid2 11324 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  x.  ,  F
) `  N )  =  (  seq  M (  x.  ,  F ) `
 n ) )
5958eqcomd 2409 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  x.  ,  F
) `  n )  =  (  seq  M (  x.  ,  F ) `
 N ) )
601, 4, 6, 27, 59climconst 12292 1  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  (  seq 
M (  x.  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    C_ wss 3280   ifcif 3699   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278    ~~> cli 12233
This theorem is referenced by:  prodmolem2a  25213
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
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-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-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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  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-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237
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