Theorem prodfap0 11556

Description: The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)

Hypotheses

Ref	Expression
prodfap0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
prodfap0.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
prodfap0.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )

Assertion

Ref	Expression
prodfap0	|- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )

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

Proof of Theorem prodfap0

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

Step	Hyp	Ref	Expression
1		prodfap0.1	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzfz2 10035	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5517	. . . . 5 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
5	4	breq1d 4015	. . . 4 |- ( m = M -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` M ) # 0 ) )
6	5	imbi2d 230	. . 3 |- ( m = M -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` M ) # 0 ) ) )
7		fveq2 5517	. . . . 5 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
8	7	breq1d 4015	. . . 4 |- ( m = n -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` n ) # 0 ) )
9	8	imbi2d 230	. . 3 |- ( m = n -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` n ) # 0 ) ) )
10		fveq2 5517	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
11	10	breq1d 4015	. . . 4 |- ( m = ( n + 1 ) -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 ) )
12	11	imbi2d 230	. . 3 |- ( m = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 ) ) )
13		fveq2 5517	. . . . 5 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
14	13	breq1d 4015	. . . 4 |- ( m = N -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` N ) # 0 ) )
15	14	imbi2d 230	. . 3 |- ( m = N -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` N ) # 0 ) ) )
16		eluzfz1 10034	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
17		elfzelz 10028	. . . . . . . 8 |- ( M e. ( M ... N ) -> M e. ZZ )
18	17	adantl 277	. . . . . . 7 |- ( ( ph /\ M e. ( M ... N ) ) -> M e. ZZ )
19		prodfap0.2	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
20	19	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ M e. ( M ... N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
21		mulcl 7941	. . . . . . . 8 |- ( ( k e. CC /\ v e. CC ) -> ( k x. v ) e. CC )
22	21	adantl 277	. . . . . . 7 |- ( ( ( ph /\ M e. ( M ... N ) ) /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
23	18, 20, 22	seq3-1 10463	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
24		fveq2 5517	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
25	24	breq1d 4015	. . . . . . . . 9 |- ( k = M -> ( ( F ` k ) # 0 <-> ( F ` M ) # 0 ) )
26	25	imbi2d 230	. . . . . . . 8 |- ( k = M -> ( ( ph -> ( F ` k ) # 0 ) <-> ( ph -> ( F ` M ) # 0 ) ) )
27		prodfap0.3	. . . . . . . . 9 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )
28	27	expcom 116	. . . . . . . 8 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) # 0 ) )
29	26, 28	vtoclga 2805	. . . . . . 7 |- ( M e. ( M ... N ) -> ( ph -> ( F ` M ) # 0 ) )
30	29	impcom 125	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( F ` M ) # 0 )
31	23, 30	eqbrtrd 4027	. . . . 5 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) # 0 )
32	31	expcom 116	. . . 4 |- ( M e. ( M ... N ) -> ( ph -> ( seq M ( x. , F ) ` M ) # 0 ) )
33	16, 32	syl 14	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( x. , F ) ` M ) # 0 ) )
34		elfzouz 10154	. . . . . . . . 9 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
35	34	3ad2ant2 1019	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> n e. ( ZZ>= ` M ) )
36	19	3ad2antl1 1159	. . . . . . . 8 |- ( ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
37	21	adantl 277	. . . . . . . 8 |- ( ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
38	35, 36, 37	seq3p1 10465	. . . . . . 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 ) ) ) )
39		elfzofz 10165	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> n e. ( M ... N ) )
40		elfzuz 10024	. . . . . . . . . . 11 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
41		eqid 2177	. . . . . . . . . . . . 13 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 |- ( ph -> M e. ZZ )
43	41, 42, 19	prodf 11549	. . . . . . . . . . . 12 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
44	43	ffvelcdmda 5654	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
45	40, 44	sylan2 286	. . . . . . . . . 10 |- ( ( ph /\ n e. ( M ... N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
46	39, 45	sylan2 286	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
47	46	3adant3 1017	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) e. CC )
48		fzofzp1 10230	. . . . . . . . . . 11 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5517	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
50	49	eleq1d 2246	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
51	50	imbi2d 230	. . . . . . . . . . . 12 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) e. CC ) <-> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) )
52		elfzuz 10024	. . . . . . . . . . . . 13 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
53	19	expcom 116	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` M ) -> ( ph -> ( F ` k ) e. CC ) )
54	52, 53	syl 14	. . . . . . . . . . . 12 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) e. CC ) )
55	51, 54	vtoclga 2805	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
56	48, 55	syl 14	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
57	56	impcom 125	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) e. CC )
58	57	3adant3 1017	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) e. CC )
59		simp3 999	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) # 0 )
60	49	breq1d 4015	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( ( F ` k ) # 0 <-> ( F ` ( n + 1 ) ) # 0 ) )
61	60	imbi2d 230	. . . . . . . . . . . 12 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) # 0 ) <-> ( ph -> ( F ` ( n + 1 ) ) # 0 ) ) )
62	61, 28	vtoclga 2805	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) # 0 ) )
63	62	impcom 125	. . . . . . . . . 10 |- ( ( ph /\ ( n + 1 ) e. ( M ... N ) ) -> ( F ` ( n + 1 ) ) # 0 )
64	48, 63	sylan2 286	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) # 0 )
65	64	3adant3 1017	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) # 0 )
66	47, 58, 59, 65	mulap0d 8618	. . . . . . 7 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) # 0 )
67	38, 66	eqbrtrd 4027	. . . . . 6 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 )
68	67	3exp 1202	. . . . 5 |- ( ph -> ( n e. ( M..^ N ) -> ( ( seq M ( x. , F ) ` n ) # 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 ) ) )
69	68	com12 30	. . . 4 |- ( n e. ( M..^ N ) -> ( ph -> ( ( seq M ( x. , F ) ` n ) # 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 ) ) )
70	69	a2d 26	. . 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, 33, 70	fzind2 10242	. 2 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( x. , F ) ` N ) # 0 ) )
72	3, 71	mpcom 36	1 |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   0cc0 7814   1c1 7815   + caddc 7817   x. cmul 7819   # cap 8541   ZZcz 9256   ZZ>=cuz 9531   ...cfz 10011  ..^cfzo 10145   seqcseq 10448

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-fz 10012  df-fzo 10146  df-seqfrec 10449

This theorem is referenced by:  prodfrecap  11557  prodfdivap  11558