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Theorem fproddccvg 11464
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 2157 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 prodrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 9443 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 14 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 10341 . . 3  |-  seq M
(  x.  ,  F
)  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
_V )
7 eqid 2157 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 9439 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 9443 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1110adantl 275 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ZZ )
12 iftrue 3510 . . . . . . . . 9  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
1312adantl 275 . . . . . . . 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 469 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
1613, 15eqeltrd 2234 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
17 iffalse 3513 . . . . . . . . 9  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
18 ax-1cn 7820 . . . . . . . . 9  |-  1  e.  CC
1917, 18eqeltrdi 2248 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
2019adantl 275 . . . . . . 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 822 . . . . . . . 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 788 . . . . . 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 5550 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
2711, 24, 26syl2anc 409 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  1 ) )
2827, 24eqeltrd 2234 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
297, 9, 28prodf 11430 . . 3  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
3029, 2ffvelrnd 5602 . 2  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
31 mulid1 7870 . . . . 5  |-  ( m  e.  CC  ->  (
m  x.  1 )  =  m )
3231adantl 275 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  CC )  ->  ( m  x.  1 )  =  m )
332adantr 274 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
34 simpr 109 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
359adantr 274 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
3628adantlr 469 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
377, 35, 36prodf 11430 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M
) --> CC )
3837, 33ffvelrnd 5602 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  x.  ,  F ) `  N
)  e.  CC )
39 elfzuz 9919 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
40 eluzelz 9443 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  ( N  +  1 ) )  ->  m  e.  ZZ )
4140adantl 275 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ZZ )
42 fprodcvg.4 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  ( M ... N ) )
4342sseld 3127 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
44 fznuz 9999 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
4543, 44syl6 33 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
4645con2d 614 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4746imp 123 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4841, 47eldifd 3112 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
49 fveqeq2 5476 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  1  <->  ( F `  m )  =  1 ) )
50 eldifi 3229 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
51 eldifn 3230 . . . . . . . . . . . 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 2248 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
5450, 53, 26syl2anc 409 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
5554, 52eqtrd 2190 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
5649, 55vtoclga 2778 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  1 )
5748, 56syl 14 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  1 )
5839, 57sylan2 284 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  1 )
5958adantlr 469 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  1 )
60 fveq2 5467 . . . . . 6  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
6160eleq1d 2226 . . . . 5  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
6228ralrimiva 2530 . . . . . 6  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
6362ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
64 simpr 109 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
6561, 63, 64rspcdva 2821 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  ( F `  m )  e.  CC )
66 mulcl 7854 . . . . 5  |-  ( ( m  e.  CC  /\  v  e.  CC )  ->  ( m  x.  v
)  e.  CC )
6766adantl 275 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  ( m  e.  CC  /\  v  e.  CC ) )  -> 
( m  x.  v
)  e.  CC )
6832, 33, 34, 38, 59, 65, 67seq3id2 10403 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  x.  ,  F ) `  N
)  =  (  seq M (  x.  ,  F ) `  n
) )
6968eqcomd 2163 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  x.  ,  F ) `  n
)  =  (  seq M (  x.  ,  F ) `  N
) )
701, 4, 6, 30, 69climconst 11182 1  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  (  seq M (  x.  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698  DECID wdc 820    = wceq 1335    e. wcel 2128   A.wral 2435   _Vcvv 2712    \ cdif 3099    C_ wss 3102   ifcif 3505   class class class wbr 3965    |-> cmpt 4025   ` cfv 5169  (class class class)co 5821   CCcc 7725   1c1 7728    + caddc 7730    x. cmul 7732   ZZcz 9162   ZZ>=cuz 9434   ...cfz 9907    seqcseq 10339    ~~> cli 11170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7818  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-icn 7822  ax-addcl 7823  ax-addrcl 7824  ax-mulcl 7825  ax-mulrcl 7826  ax-addcom 7827  ax-mulcom 7828  ax-addass 7829  ax-mulass 7830  ax-distr 7831  ax-i2m1 7832  ax-0lt1 7833  ax-1rid 7834  ax-0id 7835  ax-rnegex 7836  ax-precex 7837  ax-cnre 7838  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841  ax-pre-apti 7842  ax-pre-ltadd 7843  ax-pre-mulgt0 7844  ax-pre-mulext 7845
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-sub 8043  df-neg 8044  df-reap 8445  df-ap 8452  df-div 8541  df-inn 8829  df-2 8887  df-n0 9086  df-z 9163  df-uz 9435  df-rp 9556  df-fz 9908  df-seqfrec 10340  df-exp 10414  df-cj 10737  df-rsqrt 10893  df-abs 10894  df-clim 11171
This theorem is referenced by:  prodmodclem2a  11468
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