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Theorem fprodntrivap 11591

Description: A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)

**Hypotheses**
Ref	Expression
fprodntriv.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
fprodntriv.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
fprodntriv.3	⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))

**Assertion**
Ref	Expression
fprodntrivap	⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))

Distinct variable groups: 𝐴,𝑘,𝑛,𝑦 𝐵,𝑛,𝑦 𝑛,𝑁,𝑦 𝑘,𝑍,𝑛,𝑦 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑦,𝑘) 𝐵(𝑘) 𝑀(𝑦,𝑘,𝑛) 𝑁(𝑘)

Proof of Theorem fprodntrivap Dummy variables 𝑚 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodntriv.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		fprodntriv.1	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	1, 2	eleqtrdi 2270	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		peano2uz 9582	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
6	5, 2	eleqtrrdi 2271	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍)
7		1ap0 8546	. . 3 ⊢ 1 # 0
8		eqid 2177	. . . 4 ⊢ (ℤ_≥‘(𝑁 + 1)) = (ℤ_≥‘(𝑁 + 1))
9		eluzelz 9536	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
10	9, 2	eleq2s 2272	. . . . . 6 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
11	1, 10	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	peano2zd 9377	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
13		seqex 10446	. . . . 5 ⊢ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ∈ V
14	13	a1i 9	. . . 4 ⊢ (𝜑 → seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ∈ V)
15		1cnd 7972	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
16		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ_≥‘(𝑁 + 1)))
17		fprodntriv.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))
18	17	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝐴 ⊆ (𝑀...𝑁))
19	11	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 ∈ ℤ)
20	19	zred 9374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 ∈ ℝ)
21	19	peano2zd 9377	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ ℤ)
22	21	zred 9374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ ℝ)
23		elfzelz 10024	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ((𝑁 + 1)...𝑛) → 𝑚 ∈ ℤ)
24	23	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ ℤ)
25	24	zred 9374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ ℝ)
26	20	ltp1d 8886	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 < (𝑁 + 1))
27		elfzle1 10026	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ((𝑁 + 1)...𝑛) → (𝑁 + 1) ≤ 𝑚)
28	27	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ≤ 𝑚)
29	20, 22, 25, 26, 28	ltletrd 8379	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 < 𝑚)
30		zltnle 9298	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑁 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑁))
31	19, 24, 30	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑁))
32	29, 31	mpbid 147	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ≤ 𝑁)
33	32	intnand 931	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁))
34	33	intnand 931	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)))
35		elfz2 10014	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)))
36	34, 35	sylnibr 677	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ∈ (𝑀...𝑁))
37	18, 36	ssneldd 3158	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ∈ 𝐴)
38	37	iffalsed 3544	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) = 1)
39	6	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ 𝑍)
40		elfzuz 10020	. . . . . . . . . 10 ⊢ (𝑚 ∈ ((𝑁 + 1)...𝑛) → 𝑚 ∈ (ℤ_≥‘(𝑁 + 1)))
41	40	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ (ℤ_≥‘(𝑁 + 1)))
42	2	uztrn2 9544	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑚 ∈ 𝑍)
43	39, 41, 42	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ 𝑍)
44		ax-1cn 7903	. . . . . . . . 9 ⊢ 1 ∈ ℂ
45	38, 44	eqeltrdi 2268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ)
46		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑚
47		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐴
48		nfcsb1v 3090	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
49		nfcv 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
50	47, 48, 49	nfif 3562	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)
51		eleq1w 2238	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
52		csbeq1a 3066	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
53	51, 52	ifbieq1d 3556	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1))
54		eqid 2177	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
55	46, 50, 53, 54	fvmptf 5608	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑚) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1))
56	43, 45, 55	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑚) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1))
57		1ex 7951	. . . . . . . . 9 ⊢ 1 ∈ V
58	57	fvconst2 5732	. . . . . . . 8 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑁 + 1)) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑚) = 1)
59	41, 58	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑚) = 1)
60	38, 56, 59	3eqtr4d 2220	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑚) = (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑚))
61	6	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (𝑁 + 1) ∈ 𝑍)
62		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ (ℤ_≥‘(𝑁 + 1)))
63	2	uztrn2 9544	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ 𝑍)
64	61, 62, 63	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ 𝑍)
65	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝐴 ⊆ (𝑀...𝑁))
66	11	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 ∈ ℤ)
67	66	zred 9374	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 ∈ ℝ)
68		peano2re 8092	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
69	67, 68	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (𝑁 + 1) ∈ ℝ)
70		eluzelz 9536	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (ℤ_≥‘(𝑁 + 1)) → 𝑝 ∈ ℤ)
71	70	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ ℤ)
72	71	zred 9374	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ ℝ)
73	67	ltp1d 8886	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 < (𝑁 + 1))
74		eluzle 9539	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑝)
75	74	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (𝑁 + 1) ≤ 𝑝)
76	67, 69, 72, 73, 75	ltletrd 8379	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 < 𝑝)
77		zltnle 9298	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑁 < 𝑝 ↔ ¬ 𝑝 ≤ 𝑁))
78	66, 71, 77	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (𝑁 < 𝑝 ↔ ¬ 𝑝 ≤ 𝑁))
79	76, 78	mpbid 147	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ≤ 𝑁)
80	79	intnand 931	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁))
81	80	intnand 931	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁)))
82		elfz2 10014	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁)))
83	81, 82	sylnibr 677	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ∈ (𝑀...𝑁))
84	65, 83	ssneldd 3158	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ∈ 𝐴)
85	84	iffalsed 3544	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) = 1)
86	85, 44	eqeltrdi 2268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ)
87		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑝
88		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑝 ∈ 𝐴
89		nfcsb1v 3090	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵
90	88, 89, 49	nfif 3562	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)
91		eleq1w 2238	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → (𝑘 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴))
92		csbeq1a 3066	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → 𝐵 = ⦋𝑝 / 𝑘⦌𝐵)
93	91, 92	ifbieq1d 3556	. . . . . . . . 9 ⊢ (𝑘 = 𝑝 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
94	87, 90, 93, 54	fvmptf 5608	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑍 ∧ if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
95	64, 86, 94	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
96	95, 86	eqeltrd 2254	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) ∈ ℂ)
97	57	fvconst2 5732	. . . . . . . 8 ⊢ (𝑝 ∈ (ℤ_≥‘(𝑁 + 1)) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑝) = 1)
98	97	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑝) = 1)
99	98, 44	eqeltrdi 2268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑝) ∈ ℂ)
100		mulcl 7937	. . . . . . 7 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝 · 𝑞) ∈ ℂ)
101	100	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ)
102	16, 60, 96, 99, 101	seq3fveq 10470	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑛) = (seq(𝑁 + 1)( · , ((ℤ_≥‘(𝑁 + 1)) × {1}))‘𝑛))
103	8	prodf1 11549	. . . . . 6 ⊢ (𝑛 ∈ (ℤ_≥‘(𝑁 + 1)) → (seq(𝑁 + 1)( · , ((ℤ_≥‘(𝑁 + 1)) × {1}))‘𝑛) = 1)
104	103	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , ((ℤ_≥‘(𝑁 + 1)) × {1}))‘𝑛) = 1)
105	102, 104	eqtrd 2210	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑛) = 1)
106	8, 12, 14, 15, 105	climconst 11297	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 1)
107		breq1 4006	. . . . 5 ⊢ (𝑦 = 1 → (𝑦 # 0 ↔ 1 # 0))
108		breq2 4007	. . . . 5 ⊢ (𝑦 = 1 → (seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 1))
109	107, 108	anbi12d 473	. . . 4 ⊢ (𝑦 = 1 → ((𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (1 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 1)))
110	57, 109	spcev 2832	. . 3 ⊢ ((1 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 1) → ∃𝑦(𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
111	7, 106, 110	sylancr 414	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
112		seqeq1 10447	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))))
113	112	breq1d 4013	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
114	113	anbi2d 464	. . . 4 ⊢ (𝑛 = (𝑁 + 1) → ((𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
115	114	exbidv 1825	. . 3 ⊢ (𝑛 = (𝑁 + 1) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
116	115	rspcev 2841	. 2 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ ∃𝑦(𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)) → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
117	6, 111, 116	syl2anc 411	1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 Vcvv 2737 ⦋csb 3057 ⊆ wss 3129 ifcif 3534 {csn 3592 class class class wbr 4003 ↦ cmpt 4064 × cxp 4624 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 + caddc 7813 · cmul 7815 < clt 7991 ≤ cle 7992 # cap 8537 ℤcz 9252 ℤ_≥cuz 9527 ...cfz 10007 seqcseq 10444 ⇝ cli 11285

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-n0 9176 df-z 9253 df-uz 9528 df-rp 9653 df-fz 10008 df-fzo 10142 df-seqfrec 10445 df-exp 10519 df-cj 10850 df-rsqrt 11006 df-abs 11007 df-clim 11286

This theorem is referenced by: fprodssdc 11597

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