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Theorem fprodntrivap 12144

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 2324	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		peano2uz 9816	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
6	5, 2	eleqtrrdi 2325	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍)
7		1ap0 8769	. . 3 ⊢ 1 # 0
8		eqid 2231	. . . 4 ⊢ (ℤ_≥‘(𝑁 + 1)) = (ℤ_≥‘(𝑁 + 1))
9		eluzelz 9764	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
10	9, 2	eleq2s 2326	. . . . . 6 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
11	1, 10	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	peano2zd 9604	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
13		seqex 10710	. . . . 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 8194	. . . 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 9601	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 ∈ ℝ)
21	19	peano2zd 9604	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ ℤ)
22	21	zred 9601	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ ℝ)
23		elfzelz 10259	. . . . . . . . . . . . . . . 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 9601	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ ℝ)
26	20	ltp1d 9109	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 < (𝑁 + 1))
27		elfzle1 10261	. . . . . . . . . . . . . . 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 8602	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 < 𝑚)
30		zltnle 9524	. . . . . . . . . . . . . 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 938	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁))
34	33	intnand 938	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)))
35		elfz2 10249	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)))
36	34, 35	sylnibr 683	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ∈ (𝑀...𝑁))
37	18, 36	ssneldd 3230	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ∈ 𝐴)
38	37	iffalsed 3615	. . . . . . 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 10255	. . . . . . . . . 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 9773	. . . . . . . . 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 8124	. . . . . . . . 9 ⊢ 1 ∈ ℂ
45	38, 44	eqeltrdi 2322	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ)
46		nfcv 2374	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑚
47		nfv 1576	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐴
48		nfcsb1v 3160	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
49		nfcv 2374	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
50	47, 48, 49	nfif 3634	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)
51		eleq1w 2292	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
52		csbeq1a 3136	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
53	51, 52	ifbieq1d 3628	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1))
54		eqid 2231	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
55	46, 50, 53, 54	fvmptf 5739	. . . . . . . 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 8173	. . . . . . . . 9 ⊢ 1 ∈ V
58	57	fvconst2 5869	. . . . . . . 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 2274	. . . . . 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 9773	. . . . . . . . 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 9601	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 ∈ ℝ)
68		peano2re 8314	. . . . . . . . . . . . . . . . 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 9764	. . . . . . . . . . . . . . . . . 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 9601	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ ℝ)
73	67	ltp1d 9109	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 < (𝑁 + 1))
74		eluzle 9767	. . . . . . . . . . . . . . . . 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 8602	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 < 𝑝)
77		zltnle 9524	. . . . . . . . . . . . . . . 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 938	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁))
81	80	intnand 938	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁)))
82		elfz2 10249	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁)))
83	81, 82	sylnibr 683	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ∈ (𝑀...𝑁))
84	65, 83	ssneldd 3230	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ∈ 𝐴)
85	84	iffalsed 3615	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) = 1)
86	85, 44	eqeltrdi 2322	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ)
87		nfcv 2374	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑝
88		nfv 1576	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑝 ∈ 𝐴
89		nfcsb1v 3160	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵
90	88, 89, 49	nfif 3634	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)
91		eleq1w 2292	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → (𝑘 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴))
92		csbeq1a 3136	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → 𝐵 = ⦋𝑝 / 𝑘⦌𝐵)
93	91, 92	ifbieq1d 3628	. . . . . . . . 9 ⊢ (𝑘 = 𝑝 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
94	87, 90, 93, 54	fvmptf 5739	. . . . . . . 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 2308	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) ∈ ℂ)
97	57	fvconst2 5869	. . . . . . . 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 2322	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑝) ∈ ℂ)
100		mulcl 8158	. . . . . . 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 10740	. . . . 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 12102	. . . . . 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 2264	. . . 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 11850	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 1)
107		breq1 4091	. . . . 5 ⊢ (𝑦 = 1 → (𝑦 # 0 ↔ 1 # 0))
108		breq2 4092	. . . . 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 2901	. . 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 10711	. . . . . 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 4098	. . . . 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 1873	. . 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 2910	. 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 1004 = wceq 1397 ∃wex 1540 ∈ wcel 2202 ∃wrex 2511 Vcvv 2802 ⦋csb 3127 ⊆ wss 3200 ifcif 3605 {csn 3669 class class class wbr 4088 ↦ cmpt 4150 × cxp 4723 ‘cfv 5326 (class class class)co 6017 ℂcc 8029 ℝcr 8030 0cc0 8031 1c1 8032 + caddc 8034 · cmul 8036 < clt 8213 ≤ cle 8214 # cap 8760 ℤcz 9478 ℤ_≥cuz 9754 ...cfz 10242 seqcseq 10708 ⇝ cli 11838

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 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149

This theorem depends on definitions: df-bi 117 df-dc 842 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-rmo 2518 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-if 3606 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 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-n0 9402 df-z 9479 df-uz 9755 df-rp 9888 df-fz 10243 df-fzo 10377 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-rsqrt 11558 df-abs 11559 df-clim 11839

This theorem is referenced by: fprodssdc 12150

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