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Theorem fproddccvg 11580
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 )
prodrbdc.dc  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
prodrb.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fprodcvg.4  |-  ( ph  ->  A  C_  ( M ... N ) )
Assertion
Ref Expression
fproddccvg  |-  ( 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 fproddccvg
Dummy variables  n  v  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 prodrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 9537 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 14 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 10447 . . 3  |-  seq M
(  x.  ,  F
)  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
_V )
7 eqid 2177 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 9533 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 9537 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1110adantl 277 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ZZ )
12 iftrue 3540 . . . . . . . . 9  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
1312adantl 277 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  =  B )
14 prodmo.2 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
1514adantlr 477 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
1613, 15eqeltrd 2254 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
17 iffalse 3543 . . . . . . . . 9  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
18 ax-1cn 7904 . . . . . . . . 9  |-  1  e.  CC
1917, 18eqeltrdi 2268 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
2019adantl 277 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  -.  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
21 prodrbdc.dc . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
22 exmiddc 836 . . . . . . . 8  |-  (DECID  k  e.  A  ->  ( k  e.  A  \/  -.  k  e.  A )
)
2321, 22syl 14 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  A  \/  -.  k  e.  A )
)
2416, 20, 23mpjaodan 798 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  1 )  e.  CC )
25 prodmo.1 . . . . . . 7  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
2625fvmpt2 5600 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
2711, 24, 26syl2anc 411 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  1 ) )
2827, 24eqeltrd 2254 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
297, 9, 28prodf 11546 . . 3  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
3029, 2ffvelcdmd 5653 . 2  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
31 mulrid 7954 . . . . 5  |-  ( m  e.  CC  ->  (
m  x.  1 )  =  m )
3231adantl 277 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  CC )  ->  ( m  x.  1 )  =  m )
332adantr 276 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
34 simpr 110 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
359adantr 276 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
3628adantlr 477 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
377, 35, 36prodf 11546 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M
) --> CC )
3837, 33ffvelcdmd 5653 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  x.  ,  F ) `  N
)  e.  CC )
39 elfzuz 10021 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
40 eluzelz 9537 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  ( N  +  1 ) )  ->  m  e.  ZZ )
4140adantl 277 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ZZ )
42 fprodcvg.4 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  ( M ... N ) )
4342sseld 3155 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
44 fznuz 10102 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
4543, 44syl6 33 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
4645con2d 624 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4746imp 124 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4841, 47eldifd 3140 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
49 fveqeq2 5525 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  1  <->  ( F `  m )  =  1 ) )
50 eldifi 3258 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
51 eldifn 3259 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
5251, 17syl 14 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
5352, 18eqeltrdi 2268 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
5450, 53, 26syl2anc 411 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
5554, 52eqtrd 2210 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
5649, 55vtoclga 2804 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  1 )
5748, 56syl 14 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  1 )
5839, 57sylan2 286 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  1 )
5958adantlr 477 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  1 )
60 fveq2 5516 . . . . . 6  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
6160eleq1d 2246 . . . . 5  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
6228ralrimiva 2550 . . . . . 6  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
6362ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
64 simpr 110 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
6561, 63, 64rspcdva 2847 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  ( F `  m )  e.  CC )
66 mulcl 7938 . . . . 5  |-  ( ( m  e.  CC  /\  v  e.  CC )  ->  ( m  x.  v
)  e.  CC )
6766adantl 277 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  ( m  e.  CC  /\  v  e.  CC ) )  -> 
( m  x.  v
)  e.  CC )
6832, 33, 34, 38, 59, 65, 67seq3id2 10509 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  x.  ,  F ) `  N
)  =  (  seq M (  x.  ,  F ) `  n
) )
6968eqcomd 2183 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  x.  ,  F ) `  n
)  =  (  seq M (  x.  ,  F ) `  N
) )
701, 4, 6, 30, 69climconst 11298 1  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  (  seq M (  x.  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2738    \ cdif 3127    C_ wss 3130   ifcif 3535   class class class wbr 4004    |-> cmpt 4065   ` cfv 5217  (class class class)co 5875   CCcc 7809   1c1 7812    + caddc 7814    x. cmul 7816   ZZcz 9253   ZZ>=cuz 9528   ...cfz 10008    seqcseq 10445    ~~> cli 11286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-fz 10009  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-rsqrt 11007  df-abs 11008  df-clim 11287
This theorem is referenced by:  prodmodclem2a  11584
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