Theorem prodfap0 11710

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 10107	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5558	. . . . 5 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
5	4	breq1d 4043	. . . 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 5558	. . . . 5 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
8	7	breq1d 4043	. . . 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 5558	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
11	10	breq1d 4043	. . . 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 5558	. . . . 5 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
14	13	breq1d 4043	. . . 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 10106	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
17		elfzelz 10100	. . . . . . . 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 8006	. . . . . . . 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 10554	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
24		fveq2 5558	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
25	24	breq1d 4043	. . . . . . . . 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 2830	. . . . . . 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 4055	. . . . 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 10226	. . . . . . . . 9 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
35	34	3ad2ant2 1021	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> n e. ( ZZ>= ` M ) )
36	19	3ad2antl1 1161	. . . . . . . 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 10557	. . . . . . 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 10238	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> n e. ( M ... N ) )
40		elfzuz 10096	. . . . . . . . . . 11 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
41		eqid 2196	. . . . . . . . . . . . 13 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 |- ( ph -> M e. ZZ )
43	41, 42, 19	prodf 11703	. . . . . . . . . . . 12 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
44	43	ffvelcdmda 5697	. . . . . . . . . . 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 1019	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) e. CC )
48		fzofzp1 10303	. . . . . . . . . . 11 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5558	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
50	49	eleq1d 2265	. . . . . . . . . . . . 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 10096	. . . . . . . . . . . . 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 2830	. . . . . . . . . . 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 1019	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) e. CC )
59		simp3 1001	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) # 0 )
60	49	breq1d 4043	. . . . . . . . . . . . 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 2830	. . . . . . . . . . 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 1019	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) # 0 )
66	47, 58, 59, 65	mulap0d 8685	. . . . . . 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 4055	. . . . . 6 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 )
68	67	3exp 1204	. . . . 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 10315	. 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 980   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   # cap 8608   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083  ..^cfzo 10217   seqcseq 10539

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540

This theorem is referenced by:  prodfrecap  11711  prodfdivap  11712