Theorem prodfap0 11727

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	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
prodfap0.2	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
prodfap0.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)

Assertion

Ref	Expression
prodfap0	⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)

Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,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 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10124	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5561	. . . . 5 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
5	4	breq1d 4044	. . . 4 ⊢ (𝑚 = 𝑀 → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘𝑀) # 0))
6	5	imbi2d 230	. . 3 ⊢ (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) # 0)))
7		fveq2 5561	. . . . 5 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
8	7	breq1d 4044	. . . 4 ⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) # 0))
9	8	imbi2d 230	. . 3 ⊢ (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑛) # 0)))
10		fveq2 5561	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
11	10	breq1d 4044	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0))
12	11	imbi2d 230	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
13		fveq2 5561	. . . . 5 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
14	13	breq1d 4044	. . . 4 ⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) # 0))
15	14	imbi2d 230	. . 3 ⊢ (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)))
16		eluzfz1 10123	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
17		elfzelz 10117	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
18	17	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
19		prodfap0.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
20	19	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
21		mulcl 8023	. . . . . . . 8 ⊢ ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 · 𝑣) ∈ ℂ)
22	21	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
23	18, 20, 22	seq3-1 10571	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
24		fveq2 5561	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
25	24	breq1d 4044	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) # 0 ↔ (𝐹‘𝑀) # 0))
26	25	imbi2d 230	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝜑 → (𝐹‘𝑘) # 0) ↔ (𝜑 → (𝐹‘𝑀) # 0)))
27		prodfap0.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)
28	27	expcom 116	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) # 0))
29	26, 28	vtoclga 2830	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑀) # 0))
30	29	impcom 125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (𝐹‘𝑀) # 0)
31	23, 30	eqbrtrd 4056	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) # 0)
32	31	expcom 116	. . . 4 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) # 0))
33	16, 32	syl 14	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) # 0))
34		elfzouz 10243	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
35	34	3ad2ant2 1021	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → 𝑛 ∈ (ℤ≥‘𝑀))
36	19	3ad2antl1 1161	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
37	21	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
38	35, 36, 37	seq3p1 10574	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
39		elfzofz 10255	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (𝑀...𝑁))
40		elfzuz 10113	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
41		eqid 2196	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
43	41, 42, 19	prodf 11720	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ)
44	43	ffvelcdmda 5700	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
45	40, 44	sylan2 286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
46	39, 45	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
47	46	3adant3 1019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
48		fzofzp1 10320	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
49		fveq2 5561	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
50	49	eleq1d 2265	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
51	50	imbi2d 230	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
52		elfzuz 10113	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
53	19	expcom 116	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝐹‘𝑘) ∈ ℂ))
54	52, 53	syl 14	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ∈ ℂ))
55	51, 54	vtoclga 2830	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
56	48, 55	syl 14	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
57	56	impcom 125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
58	57	3adant3 1019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
59		simp3 1001	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) # 0)
60	49	breq1d 4044	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) # 0 ↔ (𝐹‘(𝑛 + 1)) # 0))
61	60	imbi2d 230	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) # 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) # 0)))
62	61, 28	vtoclga 2830	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) # 0))
63	62	impcom 125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) # 0)
64	48, 63	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) # 0)
65	64	3adant3 1019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) # 0)
66	47, 58, 59, 65	mulap0d 8702	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) # 0)
67	38, 66	eqbrtrd 4056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)
68	67	3exp 1204	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐹)‘𝑛) # 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
69	68	com12 30	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) # 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
70	69	a2d 26	. . 3 ⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
71	6, 9, 12, 15, 33, 70	fzind2 10332	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0))
72	3, 71	mpcom 36	1 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   class class class wbr 4034  ‘cfv 5259  (class class class)co 5925  ℂcc 7894  0cc0 7896  1c1 7897   + caddc 7899   · cmul 7901   # cap 8625  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  ..^cfzo 10234  seqcseq 10556

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-fzo 10235  df-seqfrec 10557

This theorem is referenced by:  prodfrecap  11728  prodfdivap  11729