Theorem prodfap0 12108

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 10267	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5639	. . . . 5 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
5	4	breq1d 4098	. . . 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 5639	. . . . 5 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
8	7	breq1d 4098	. . . 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 5639	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
11	10	breq1d 4098	. . . 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 5639	. . . . 5 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
14	13	breq1d 4098	. . . 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 10266	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
17		elfzelz 10260	. . . . . . . 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 8159	. . . . . . . 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 10725	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
24		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
25	24	breq1d 4098	. . . . . . . . 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 2870	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑀) # 0))
30	29	impcom 125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (𝐹‘𝑀) # 0)
31	23, 30	eqbrtrd 4110	. . . . 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 10386	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
35	34	3ad2ant2 1045	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → 𝑛 ∈ (ℤ≥‘𝑀))
36	19	3ad2antl1 1185	. . . . . . . 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 10728	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
39		elfzofz 10398	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (𝑀...𝑁))
40		elfzuz 10256	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
41		eqid 2231	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
43	41, 42, 19	prodf 12101	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ)
44	43	ffvelcdmda 5782	. . . . . . . . . . 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 1043	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
48		fzofzp1 10473	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
49		fveq2 5639	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
50	49	eleq1d 2300	. . . . . . . . . . . . 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 10256	. . . . . . . . . . . . 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 2870	. . . . . . . . . . 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 1043	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
59		simp3 1025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) # 0)
60	49	breq1d 4098	. . . . . . . . . . . . 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 2870	. . . . . . . . . . 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 1043	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) # 0)
66	47, 58, 59, 65	mulap0d 8838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) # 0)
67	38, 66	eqbrtrd 4110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)
68	67	3exp 1228	. . . . 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 10486	. 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 1004   = wceq 1397   ∈ wcel 2202   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  ℂcc 8030  0cc0 8032  1c1 8033   + caddc 8035   · cmul 8037   # cap 8761  ℤcz 9479  ℤ_≥cuz 9755  ...cfz 10243  ..^cfzo 10377  seqcseq 10710

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-seqfrec 10711

This theorem is referenced by:  prodfrecap  12109  prodfdivap  12110