Theorem iprodclim3 14731

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.

Step	Hyp	Ref	Expression
1		iprodclim3.4	. . 3 |- ( ph -> F e. dom ~~> )
2		climdm 14285	. . 3 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
3	1, 2	sylib 208	. 2 |- ( ph -> F ~~> ( ~~> ` F ) )
4		prodfc 14675	. . . 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 2623	. . . . 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 2622	. . . . . . 7 |- ( k e. Z |-> A ) = ( k e. Z |-> A )
11	9, 10	fmptd 6385	. . . . . 6 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
12	11	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
13	5, 6, 7, 8, 12	iprod 14668	. . . 4 |- ( ph -> prod_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( x. , ( k e. Z |-> A ) ) ) )
14	4, 13	syl5eqr 2670	. . 3 |- ( ph -> prod_ k e. Z A = ( ~~> ` seq M ( x. , ( k e. Z |-> A ) ) ) )
15		seqex 12803	. . . . . . 7 |- seq M ( x. , ( k e. Z |-> A ) ) e. _V
16	15	a1i 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 12382	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
19	18, 5	sseqtr4i 3638	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
20		resmpt 5449	. . . . . . . . . . . . 13 |- ( ( M ... j ) C_ Z -> ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A ) )
21	19, 20	ax-mp 5	. . . . . . . . . . . 12 |- ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A )
22	21	fveq1i 6192	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
23		fvres 6207	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
24	22, 23	syl5reqr 2671	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
25	24	prodeq2i 14649	. . . . . . . . 9 |- prod_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = prod_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
26		prodfc 14675	. . . . . . . . 9 |- prod_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m ) = prod_ k e. ( M ... j ) A
27	25, 26	eqtri 2644	. . . . . . . 8 |- prod_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = prod_ k e. ( M ... j ) A
28		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ j e. Z ) /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
29		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ j e. Z ) -> j e. Z )
30	29, 5	syl6eleq 2711	. . . . . . . . 9 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
31		elfzuz 12338	. . . . . . . . . . . 12 |- ( m e. ( M ... j ) -> m e. ( ZZ>= ` M ) )
32	31, 5	syl6eleqr 2712	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> m e. Z )
33	32, 12	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
34	33	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ j e. Z ) /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
35	28, 30, 34	fprodser 14679	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> prod_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = ( seq M ( x. , ( k e. Z |-> A ) ) ` j ) )
36	27, 35	syl5eqr 2670	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> prod_ k e. ( M ... j ) A = ( seq M ( x. , ( k e. Z |-> A ) ) ` j ) )
37	17, 36	eqtr2d 2657	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
38	5, 16, 1, 6, 37	climeq 14298	. . . . 5 |- ( ph -> ( seq M ( x. , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
39	38	iotabidv 5872	. . . 4 |- ( ph -> ( iota x seq M ( x. , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
40		df-fv 5896	. . . 4 |- ( ~~> ` seq M ( x. , ( k e. Z |-> A ) ) ) = ( iota x seq M ( x. , ( k e. Z |-> A ) ) ~~> x )
41		df-fv 5896	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
42	39, 40, 41	3eqtr4g 2681	. . 3 |- ( ph -> ( ~~> ` seq M ( x. , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
43	14, 42	eqtrd 2656	. 2 |- ( ph -> prod_ k e. Z A = ( ~~> ` F ) )
44	3, 43	breqtrrd 4681	1 |- ( ph -> F ~~> prod_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   iotacio 5849   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636

This theorem is referenced by: (None)