Theorem prodfap0 12224

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 10362	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5669	. . . . 5 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
5	4	breq1d 4118	. . . 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 5669	. . . . 5 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
8	7	breq1d 4118	. . . 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 5669	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
11	10	breq1d 4118	. . . 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 5669	. . . . 5 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
14	13	breq1d 4118	. . . 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 10361	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
17		elfzelz 10355	. . . . . . . 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 8250	. . . . . . . 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 10820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
24		fveq2 5669	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
25	24	breq1d 4118	. . . . . . . . 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 2880	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑀) # 0))
30	29	impcom 125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (𝐹‘𝑀) # 0)
31	23, 30	eqbrtrd 4130	. . . . 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 10481	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
35	34	3ad2ant2 1046	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → 𝑛 ∈ (ℤ≥‘𝑀))
36	19	3ad2antl1 1186	. . . . . . . 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 10823	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
39		elfzofz 10493	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (𝑀...𝑁))
40		elfzuz 10351	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
41		eqid 2232	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
42	1, 16, 17	3syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
43	41, 42, 19	prodf 12217	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ)
44	43	ffvelcdmda 5811	. . . . . . . . . . 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 1044	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
48		fzofzp1 10568	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
49		fveq2 5669	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
50	49	eleq1d 2301	. . . . . . . . . . . . 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 10351	. . . . . . . . . . . . 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 2880	. . . . . . . . . . 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 1044	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
59		simp3 1026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) # 0)
60	49	breq1d 4118	. . . . . . . . . . . . 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 2880	. . . . . . . . . . 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 1044	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) # 0)
66	47, 58, 59, 65	mulap0d 8928	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) # 0)
67	38, 66	eqbrtrd 4130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)
68	67	3exp 1229	. . . . 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 10581	. 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 1005   = wceq 1398   ∈ wcel 2203   class class class wbr 4108  ‘cfv 5351  (class class class)co 6049  ℂcc 8121  0cc0 8123  1c1 8124   + caddc 8126   · cmul 8128   # cap 8851  ℤcz 9573  ℤ_≥cuz 9849  ...cfz 10338  ..^cfzo 10472  seqcseq 10805

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-fzo 10473  df-seqfrec 10806

This theorem is referenced by:  prodfrecap  12225  prodfdivap  12226