Theorem fproddccvg 12083

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.

Step	Hyp	Ref	Expression
1		eqid 2229	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		prodrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9731	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. 2 |- ( ph -> N e. ZZ )
5		seqex 10671	. . 3 |- seq M ( x. , F ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( x. , F ) e. _V )
7		eqid 2229	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 9727	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 14	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 9731	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 277	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 3607	. . . . . . . . 9 |- ( k e. A -> if ( k e. A , B , 1 ) = B )
13	12	adantl 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 )
15	14	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
16	13, 15	eqeltrd 2306	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> if ( k e. A , B , 1 ) e. CC )
17		iffalse 3610	. . . . . . . . 9 |- ( -. k e. A -> if ( k e. A , B , 1 ) = 1 )
18		ax-1cn 8092	. . . . . . . . 9 |- 1 e. CC
19	17, 18	eqeltrdi 2320	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 1 ) e. CC )
20	19	adantl 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 841	. . . . . . . 8 |- (DECID k e. A -> ( k e. A \/ -. k e. A ) )
23	21, 22	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A \/ -. k e. A ) )
24	16, 20, 23	mpjaodan 803	. . . . . 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 ) )
26	25	fvmpt2 5718	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 1 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
27	11, 24, 26	syl2anc 411	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
28	27, 24	eqeltrd 2306	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
29	7, 9, 28	prodf 12049	. . 3 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
30	29, 2	ffvelcdmd 5771	. 2 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
31		mulrid 8143	. . . . 5 |- ( m e. CC -> ( m x. 1 ) = m )
32	31	adantl 277	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. CC ) -> ( m x. 1 ) = m )
33	2	adantr 276	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
34		simpr 110	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> n e. ( ZZ>= ` N ) )
35	9	adantr 276	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> M e. ZZ )
36	28	adantlr 477	. . . . . 6 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
37	7, 35, 36	prodf 12049	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
38	37, 33	ffvelcdmd 5771	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
39		elfzuz 10217	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
40		eluzelz 9731	. . . . . . . . 9 |- ( m e. ( ZZ>= ` ( N + 1 ) ) -> m e. ZZ )
41	40	adantl 277	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ZZ )
42		fprodcvg.4	. . . . . . . . . . . 12 |- ( ph -> A C_ ( M ... N ) )
43	42	sseld 3223	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
44		fznuz 10298	. . . . . . . . . . 11 |- ( m e. ( M ... N ) -> -. m e. ( ZZ>= ` ( N + 1 ) ) )
45	43, 44	syl6 33	. . . . . . . . . 10 |- ( ph -> ( m e. A -> -. m e. ( ZZ>= ` ( N + 1 ) ) ) )
46	45	con2d 627	. . . . . . . . 9 |- ( ph -> ( m e. ( ZZ>= ` ( N + 1 ) ) -> -. m e. A ) )
47	46	imp 124	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> -. m e. A )
48	41, 47	eldifd 3207	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
49		fveqeq2 5636	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 1 <-> ( F ` m ) = 1 ) )
50		eldifi 3326	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
51		eldifn 3327	. . . . . . . . . . . 12 |- ( k e. ( ZZ \ A ) -> -. k e. A )
52	51, 17	syl 14	. . . . . . . . . . 11 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 1 ) = 1 )
53	52, 18	eqeltrdi 2320	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 1 ) e. CC )
54	50, 53, 26	syl2anc 411	. . . . . . . . 9 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
55	54, 52	eqtrd 2262	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 1 )
56	49, 55	vtoclga 2867	. . . . . . 7 |- ( m e. ( ZZ \ A ) -> ( F ` m ) = 1 )
57	48, 56	syl 14	. . . . . 6 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` m ) = 1 )
58	39, 57	sylan2 286	. . . . 5 |- ( ( ph /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 1 )
59	58	adantlr 477	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 1 )
60		fveq2 5627	. . . . . 6 |- ( k = m -> ( F ` k ) = ( F ` m ) )
61	60	eleq1d 2298	. . . . 5 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
62	28	ralrimiva 2603	. . . . . 6 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
63	62	ad2antrr 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 ) )
65	61, 63, 64	rspcdva 2912	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> ( F ` m ) e. CC )
66		mulcl 8126	. . . . 5 |- ( ( m e. CC /\ v e. CC ) -> ( m x. v ) e. CC )
67	66	adantl 277	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ ( m e. CC /\ v e. CC ) ) -> ( m x. v ) e. CC )
68	32, 33, 34, 38, 59, 65, 67	seq3id2 10748	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` N ) = ( seq M ( x. , F ) ` n ) )
69	68	eqcomd 2235	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` n ) = ( seq M ( x. , F ) ` N ) )
70	1, 4, 6, 30, 69	climconst 11801	1 |- ( ph -> seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   \ cdif 3194   C_ wss 3197   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   ` cfv 5318  (class class class)co 6001   CCcc 7997   1c1 8000   + caddc 8002   x. cmul 8004   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   seqcseq 10669   ~~> cli 11789

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-fz 10205  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-rsqrt 11509  df-abs 11510  df-clim 11790

This theorem is referenced by:  prodmodclem2a  12087