Theorem fprodcvg 14660

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		prodrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 11697	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 17	. 2 |- ( ph -> N e. ZZ )
5		seqex 12803	. . 3 |- seq M ( x. , F ) e. _V
6	5	a1i 11	. 2 |- ( ph -> seq M ( x. , F ) e. _V )
7		eqid 2622	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 11692	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 17	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 11697	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 482	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 4092	. . . . . . . . . 10 |- ( k e. A -> if ( k e. A , B , 1 ) = B )
13	12	adantl 482	. . . . . . . . 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 )
15	14	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
16	13, 15	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> if ( k e. A , B , 1 ) e. CC )
17	16	ex 450	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A -> if ( k e. A , B , 1 ) e. CC ) )
18		iffalse 4095	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 1 ) = 1 )
19		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
20	18, 19	syl6eqel 2709	. . . . . . 7 |- ( -. k e. A -> if ( k e. A , B , 1 ) e. CC )
21	17, 20	pm2.61d1 171	. . . . . 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 ) )
23	22	fvmpt2 6291	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 1 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
24	11, 21, 23	syl2anc 693	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
25	24, 21	eqeltrd 2701	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
26	7, 9, 25	prodf 14619	. . 3 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
27	26, 2	ffvelrnd 6360	. 2 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
28		mulid1 10037	. . . . 5 |- ( m e. CC -> ( m x. 1 ) = m )
29	28	adantl 482	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. CC ) -> ( m x. 1 ) = m )
30	2	adantr 481	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
31		simpr 477	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> n e. ( ZZ>= ` N ) )
32	9	adantr 481	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> M e. ZZ )
33	25	adantlr 751	. . . . . 6 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	7, 32, 33	prodf 14619	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
35	34, 30	ffvelrnd 6360	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
36		elfzuz 12338	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
37		eluzelz 11697	. . . . . . . . 9 |- ( m e. ( ZZ>= ` ( N + 1 ) ) -> m e. ZZ )
38	37	adantl 482	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ZZ )
39		fprodcvg.4	. . . . . . . . . . . 12 |- ( ph -> A C_ ( M ... N ) )
40	39	sseld 3602	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
41		fznuz 12422	. . . . . . . . . . 11 |- ( m e. ( M ... N ) -> -. m e. ( ZZ>= ` ( N + 1 ) ) )
42	40, 41	syl6 35	. . . . . . . . . 10 |- ( ph -> ( m e. A -> -. m e. ( ZZ>= ` ( N + 1 ) ) ) )
43	42	con2d 129	. . . . . . . . 9 |- ( ph -> ( m e. ( ZZ>= ` ( N + 1 ) ) -> -. m e. A ) )
44	43	imp 445	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> -. m e. A )
45	38, 44	eldifd 3585	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
46		fveq2 6191	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
47	46	eqeq1d 2624	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 1 <-> ( F ` m ) = 1 ) )
48		eldifi 3732	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
49		eldifn 3733	. . . . . . . . . . . 12 |- ( k e. ( ZZ \ A ) -> -. k e. A )
50	49, 18	syl 17	. . . . . . . . . . 11 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 1 ) = 1 )
51	50, 19	syl6eqel 2709	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 1 ) e. CC )
52	48, 51, 23	syl2anc 693	. . . . . . . . 9 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
53	52, 50	eqtrd 2656	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 1 )
54	47, 53	vtoclga 3272	. . . . . . 7 |- ( m e. ( ZZ \ A ) -> ( F ` m ) = 1 )
55	45, 54	syl 17	. . . . . 6 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` m ) = 1 )
56	36, 55	sylan2 491	. . . . 5 |- ( ( ph /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 1 )
57	56	adantlr 751	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 1 )
58	29, 30, 31, 35, 57	seqid2 12847	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` N ) = ( seq M ( x. , F ) ` n ) )
59	58	eqcomd 2628	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` n ) = ( seq M ( x. , F ) ` N ) )
60	1, 4, 6, 27, 59	climconst 14274	1 |- ( ph -> seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  prodmolem2a  14664