Theorem prodfap0 12312

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 10436	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5695	. . . . 5 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
5	4	breq1d 4140	. . . 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 5695	. . . . 5 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
8	7	breq1d 4140	. . . 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 5695	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
11	10	breq1d 4140	. . . 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 5695	. . . . 5 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
14	13	breq1d 4140	. . . 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 10435	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
17		elfzelz 10428	. . . . . . . 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 481	. . . . . . 7 |- ( ( ( ph /\ M e. ( M ... N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
21		mulcl 8306	. . . . . . . 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 10899	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
24		fveq2 5695	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
25	24	breq1d 4140	. . . . . . . . 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 2889	. . . . . . 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 4152	. . . . 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 10558	. . . . . . . . 9 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
35	34	3ad2ant2 1050	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> n e. ( ZZ>= ` M ) )
36	19	3ad2antl1 1190	. . . . . . . 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 10902	. . . . . . 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 10570	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> n e. ( M ... N ) )
40		elfzuz 10424	. . . . . . . . . . 11 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
41		eqid 2238	. . . . . . . . . . . . 13 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 |- ( ph -> M e. ZZ )
43	41, 42, 19	prodf 12305	. . . . . . . . . . . 12 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
44	43	ffvelcdmda 5843	. . . . . . . . . . 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 1048	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) e. CC )
48		fzofzp1 10645	. . . . . . . . . . 11 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5695	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
50	49	eleq1d 2307	. . . . . . . . . . . . 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 10424	. . . . . . . . . . . . 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 2889	. . . . . . . . . . 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 1048	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) e. CC )
59		simp3 1030	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) # 0 )
60	49	breq1d 4140	. . . . . . . . . . . . 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 2889	. . . . . . . . . . 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 1048	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) # 0 )
66	47, 58, 59, 65	mulap0d 8986	. . . . . . 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 4152	. . . . . 6 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 )
68	67	3exp 1233	. . . . 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 10658	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   # cap 8909   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411  ..^cfzo 10549   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-seqfrec 10885

This theorem is used by:  prodfrecap  12313  prodfdivap  12314