Theorem prodfn0 14626

Description: No term of a nonzero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.)

Hypotheses

Ref	Expression
prodfn0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
prodfn0.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
prodfn0.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) =/= 0 )

Assertion

Ref	Expression
prodfn0	|- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )

Distinct variable groups:   k, F   ph, k   k, M   k, N

Proof of Theorem prodfn0

Dummy variables m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfn0.1	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzfz2 12349	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 17	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 6191	. . . . 5 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
5	4	neeq1d 2853	. . . 4 |- ( m = M -> ( ( seq M ( x. , F ) ` m ) =/= 0 <-> ( seq M ( x. , F ) ` M ) =/= 0 ) )
6	5	imbi2d 330	. . 3 |- ( m = M -> ( ( ph -> ( seq M ( x. , F ) ` m ) =/= 0 ) <-> ( ph -> ( seq M ( x. , F ) ` M ) =/= 0 ) ) )
7		fveq2 6191	. . . . 5 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
8	7	neeq1d 2853	. . . 4 |- ( m = n -> ( ( seq M ( x. , F ) ` m ) =/= 0 <-> ( seq M ( x. , F ) ` n ) =/= 0 ) )
9	8	imbi2d 330	. . 3 |- ( m = n -> ( ( ph -> ( seq M ( x. , F ) ` m ) =/= 0 ) <-> ( ph -> ( seq M ( x. , F ) ` n ) =/= 0 ) ) )
10		fveq2 6191	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
11	10	neeq1d 2853	. . . 4 |- ( m = ( n + 1 ) -> ( ( seq M ( x. , F ) ` m ) =/= 0 <-> ( seq M ( x. , F ) ` ( n + 1 ) ) =/= 0 ) )
12	11	imbi2d 330	. . 3 |- ( m = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , F ) ` m ) =/= 0 ) <-> ( ph -> ( seq M ( x. , F ) ` ( n + 1 ) ) =/= 0 ) ) )
13		fveq2 6191	. . . . 5 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
14	13	neeq1d 2853	. . . 4 |- ( m = N -> ( ( seq M ( x. , F ) ` m ) =/= 0 <-> ( seq M ( x. , F ) ` N ) =/= 0 ) )
15	14	imbi2d 330	. . 3 |- ( m = N -> ( ( ph -> ( seq M ( x. , F ) ` m ) =/= 0 ) <-> ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 ) ) )
16		eluzfz1 12348	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
17		elfzelz 12342	. . . . . . . 8 |- ( M e. ( M ... N ) -> M e. ZZ )
18	17	adantl 482	. . . . . . 7 |- ( ( ph /\ M e. ( M ... N ) ) -> M e. ZZ )
19		seq1 12814	. . . . . . 7 |- ( M e. ZZ -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
20	18, 19	syl 17	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
21		fveq2 6191	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
22	21	neeq1d 2853	. . . . . . . . 9 |- ( k = M -> ( ( F ` k ) =/= 0 <-> ( F ` M ) =/= 0 ) )
23	22	imbi2d 330	. . . . . . . 8 |- ( k = M -> ( ( ph -> ( F ` k ) =/= 0 ) <-> ( ph -> ( F ` M ) =/= 0 ) ) )
24		prodfn0.3	. . . . . . . . 9 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) =/= 0 )
25	24	expcom 451	. . . . . . . 8 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) =/= 0 ) )
26	23, 25	vtoclga 3272	. . . . . . 7 |- ( M e. ( M ... N ) -> ( ph -> ( F ` M ) =/= 0 ) )
27	26	impcom 446	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( F ` M ) =/= 0 )
28	20, 27	eqnetrd 2861	. . . . 5 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) =/= 0 )
29	28	expcom 451	. . . 4 |- ( M e. ( M ... N ) -> ( ph -> ( seq M ( x. , F ) ` M ) =/= 0 ) )
30	16, 29	syl 17	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( x. , F ) ` M ) =/= 0 ) )
31		elfzouz 12474	. . . . . . . . 9 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
32	31	3ad2ant2 1083	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> n e. ( ZZ>= ` M ) )
33		seqp1 12816	. . . . . . . 8 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
34	32, 33	syl 17	. . . . . . 7 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
35		elfzofz 12485	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> n e. ( M ... N ) )
36		elfzuz 12338	. . . . . . . . . . . 12 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
37	36	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( M ... N ) ) -> n e. ( ZZ>= ` M ) )
38		elfzuz3 12339	. . . . . . . . . . . . . . 15 |- ( n e. ( M ... N ) -> N e. ( ZZ>= ` n ) )
39		fzss2 12381	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` n ) -> ( M ... n ) C_ ( M ... N ) )
40	38, 39	syl 17	. . . . . . . . . . . . . 14 |- ( n e. ( M ... N ) -> ( M ... n ) C_ ( M ... N ) )
41	40	sselda 3603	. . . . . . . . . . . . 13 |- ( ( n e. ( M ... N ) /\ k e. ( M ... n ) ) -> k e. ( M ... N ) )
42		prodfn0.2	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
43	41, 42	sylan2 491	. . . . . . . . . . . 12 |- ( ( ph /\ ( n e. ( M ... N ) /\ k e. ( M ... n ) ) ) -> ( F ` k ) e. CC )
44	43	anassrs 680	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( M ... N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
45		mulcl 10020	. . . . . . . . . . . 12 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
46	45	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( M ... N ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
47	37, 44, 46	seqcl 12821	. . . . . . . . . 10 |- ( ( ph /\ n e. ( M ... N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
48	35, 47	sylan2 491	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
49	48	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( seq M ( x. , F ) ` n ) e. CC )
50		fzofzp1 12565	. . . . . . . . . . 11 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
51		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
52	51	eleq1d 2686	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
53	52	imbi2d 330	. . . . . . . . . . . 12 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) e. CC ) <-> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) )
54	42	expcom 451	. . . . . . . . . . . 12 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) e. CC ) )
55	53, 54	vtoclga 3272	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
56	50, 55	syl 17	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
57	56	impcom 446	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) e. CC )
58	57	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( F ` ( n + 1 ) ) e. CC )
59		simp3 1063	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( seq M ( x. , F ) ` n ) =/= 0 )
60	51	neeq1d 2853	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( ( F ` k ) =/= 0 <-> ( F ` ( n + 1 ) ) =/= 0 ) )
61	60	imbi2d 330	. . . . . . . . . . . 12 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) =/= 0 ) <-> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) ) )
62	61, 25	vtoclga 3272	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) )
63	62	impcom 446	. . . . . . . . . 10 |- ( ( ph /\ ( n + 1 ) e. ( M ... N ) ) -> ( F ` ( n + 1 ) ) =/= 0 )
64	50, 63	sylan2 491	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) =/= 0 )
65	64	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( F ` ( n + 1 ) ) =/= 0 )
66	49, 58, 59, 65	mulne0d 10679	. . . . . . 7 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) =/= 0 )
67	34, 66	eqnetrd 2861	. . . . . 6 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) =/= 0 )
68	67	3exp 1264	. . . . 5 |- ( ph -> ( n e. ( M..^ N ) -> ( ( seq M ( x. , F ) ` n ) =/= 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) =/= 0 ) ) )
69	68	com12 32	. . . 4 |- ( n e. ( M..^ N ) -> ( ph -> ( ( seq M ( x. , F ) ` n ) =/= 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) =/= 0 ) ) )
70	69	a2d 29	. . 3 |- ( n e. ( M..^ N ) -> ( ( ph -> ( seq M ( x. , F ) ` n ) =/= 0 ) -> ( ph -> ( seq M ( x. , F ) ` ( n + 1 ) ) =/= 0 ) ) )
71	6, 9, 12, 15, 30, 70	fzind2 12586	. 2 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 ) )
72	3, 71	mpcom 38	1 |- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   seqcseq 12801

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802

This theorem is referenced by:  prodfrec  14627  prodfdiv  14628  fprodn0  14709