Theorem prodfap0 11486

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 9967	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5486	. . . . 5 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
5	4	breq1d 3992	. . . 4 |- ( m = M -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` M ) # 0 ) )
6	5	imbi2d 229	. . 3 |- ( m = M -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` M ) # 0 ) ) )
7		fveq2 5486	. . . . 5 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
8	7	breq1d 3992	. . . 4 |- ( m = n -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` n ) # 0 ) )
9	8	imbi2d 229	. . 3 |- ( m = n -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` n ) # 0 ) ) )
10		fveq2 5486	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
11	10	breq1d 3992	. . . 4 |- ( m = ( n + 1 ) -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 ) )
12	11	imbi2d 229	. . 3 |- ( m = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 ) ) )
13		fveq2 5486	. . . . 5 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
14	13	breq1d 3992	. . . 4 |- ( m = N -> ( ( seq M ( x. , F ) ` m ) # 0 <-> ( seq M ( x. , F ) ` N ) # 0 ) )
15	14	imbi2d 229	. . 3 |- ( m = N -> ( ( ph -> ( seq M ( x. , F ) ` m ) # 0 ) <-> ( ph -> ( seq M ( x. , F ) ` N ) # 0 ) ) )
16		eluzfz1 9966	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
17		elfzelz 9960	. . . . . . . 8 |- ( M e. ( M ... N ) -> M e. ZZ )
18	17	adantl 275	. . . . . . 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 469	. . . . . . 7 |- ( ( ( ph /\ M e. ( M ... N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
21		mulcl 7880	. . . . . . . 8 |- ( ( k e. CC /\ v e. CC ) -> ( k x. v ) e. CC )
22	21	adantl 275	. . . . . . 7 |- ( ( ( ph /\ M e. ( M ... N ) ) /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
23	18, 20, 22	seq3-1 10395	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
24		fveq2 5486	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
25	24	breq1d 3992	. . . . . . . . 9 |- ( k = M -> ( ( F ` k ) # 0 <-> ( F ` M ) # 0 ) )
26	25	imbi2d 229	. . . . . . . 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 115	. . . . . . . 8 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) # 0 ) )
29	26, 28	vtoclga 2792	. . . . . . 7 |- ( M e. ( M ... N ) -> ( ph -> ( F ` M ) # 0 ) )
30	29	impcom 124	. . . . . 6 |- ( ( ph /\ M e. ( M ... N ) ) -> ( F ` M ) # 0 )
31	23, 30	eqbrtrd 4004	. . . . 5 |- ( ( ph /\ M e. ( M ... N ) ) -> ( seq M ( x. , F ) ` M ) # 0 )
32	31	expcom 115	. . . 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 10086	. . . . . . . . 9 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
35	34	3ad2ant2 1009	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> n e. ( ZZ>= ` M ) )
36	19	3ad2antl1 1149	. . . . . . . 8 |- ( ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
37	21	adantl 275	. . . . . . . 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 10397	. . . . . . 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 10097	. . . . . . . . . 10 |- ( n e. ( M..^ N ) -> n e. ( M ... N ) )
40		elfzuz 9956	. . . . . . . . . . 11 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
41		eqid 2165	. . . . . . . . . . . . 13 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 |- ( ph -> M e. ZZ )
43	41, 42, 19	prodf 11479	. . . . . . . . . . . 12 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
44	43	ffvelrnda 5620	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
45	40, 44	sylan2 284	. . . . . . . . . 10 |- ( ( ph /\ n e. ( M ... N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
46	39, 45	sylan2 284	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
47	46	3adant3 1007	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) e. CC )
48		fzofzp1 10162	. . . . . . . . . . 11 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5486	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
50	49	eleq1d 2235	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
51	50	imbi2d 229	. . . . . . . . . . . 12 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) e. CC ) <-> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) )
52		elfzuz 9956	. . . . . . . . . . . . 13 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
53	19	expcom 115	. . . . . . . . . . . . 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 2792	. . . . . . . . . . 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 124	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) e. CC )
58	57	3adant3 1007	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) e. CC )
59		simp3 989	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` n ) # 0 )
60	49	breq1d 3992	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( ( F ` k ) # 0 <-> ( F ` ( n + 1 ) ) # 0 ) )
61	60	imbi2d 229	. . . . . . . . . . . 12 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) # 0 ) <-> ( ph -> ( F ` ( n + 1 ) ) # 0 ) ) )
62	61, 28	vtoclga 2792	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) # 0 ) )
63	62	impcom 124	. . . . . . . . . 10 |- ( ( ph /\ ( n + 1 ) e. ( M ... N ) ) -> ( F ` ( n + 1 ) ) # 0 )
64	48, 63	sylan2 284	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) # 0 )
65	64	3adant3 1007	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( F ` ( n + 1 ) ) # 0 )
66	47, 58, 59, 65	mulap0d 8555	. . . . . . 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 4004	. . . . . 6 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , F ) ` n ) # 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) # 0 )
68	67	3exp 1192	. . . . 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 10174	. 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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   # cap 8479   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944  ..^cfzo 10077   seqcseq 10380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945  df-fzo 10078  df-seqfrec 10381

This theorem is referenced by:  prodfrecap  11487  prodfdivap  11488