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Theorem fprodntrivap 12329

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 2331	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		peano2uz 9962	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
6	5, 2	eleqtrrdi 2332	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍)
7		1ap0 8908	. . 3 ⊢ 1 # 0
8		eqid 2238	. . . 4 ⊢ (ℤ_≥‘(𝑁 + 1)) = (ℤ_≥‘(𝑁 + 1))
9		eluzelz 9910	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
10	9, 2	eleq2s 2333	. . . . . 6 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
11	1, 10	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	peano2zd 9750	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
13		seqex 10864	. . . . 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 8332	. . . 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 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝐴 ⊆ (𝑀...𝑁))
19	11	ad2antrr 492	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 ∈ ℤ)
20	19	zred 9747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 ∈ ℝ)
21	19	peano2zd 9750	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ ℤ)
22	21	zred 9747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ ℝ)
23		elfzelz 10407	. . . . . . . . . . . . . . . 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 9747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ ℝ)
26	20	ltp1d 9250	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 < (𝑁 + 1))
27		elfzle1 10410	. . . . . . . . . . . . . . 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 8741	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑁 < 𝑚)
30		zltnle 9669	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑁 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑁))
31	19, 24, 30	syl2anc 415	. . . . . . . . . . . . 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 943	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁))
34	33	intnand 943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)))
35		elfz2 10397	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑀 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)))
36	34, 35	sylnibr 688	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ∈ (𝑀...𝑁))
37	18, 36	ssneldd 3251	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → ¬ 𝑚 ∈ 𝐴)
38	37	iffalsed 3647	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) = 1)
39	6	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝑁 + 1) ∈ 𝑍)
40		elfzuz 10403	. . . . . . . . . 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 9919	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑚 ∈ 𝑍)
43	39, 41, 42	syl2anc 415	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → 𝑚 ∈ 𝑍)
44		ax-1cn 8262	. . . . . . . . 9 ⊢ 1 ∈ ℂ
45	38, 44	eqeltrdi 2329	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ)
46		nfcv 2392	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑚
47		nfv 1581	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐴
48		nfcsb1v 3180	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
49		nfcv 2392	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
50	47, 48, 49	nfif 3666	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)
51		eleq1w 2299	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
52		csbeq1a 3156	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
53	51, 52	ifbieq1d 3660	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1))
54		eqid 2238	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
55	46, 50, 53, 54	fvmptf 5792	. . . . . . . 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 415	. . . . . . 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 8311	. . . . . . . . 9 ⊢ 1 ∈ V
58	57	fvconst2 5922	. . . . . . . 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 2281	. . . . . 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 492	. . . . . . . . 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 9919	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ 𝑍)
64	61, 62, 63	syl2anc 415	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ 𝑍)
65	17	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝐴 ⊆ (𝑀...𝑁))
66	11	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 ∈ ℤ)
67	66	zred 9747	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 ∈ ℝ)
68		peano2re 8452	. . . . . . . . . . . . . . . . 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 9910	. . . . . . . . . . . . . . . . . 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 9747	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑝 ∈ ℝ)
73	67	ltp1d 9250	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 < (𝑁 + 1))
74		eluzle 9913	. . . . . . . . . . . . . . . . 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 8741	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 < 𝑝)
77		zltnle 9669	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑁 < 𝑝 ↔ ¬ 𝑝 ≤ 𝑁))
78	66, 71, 77	syl2anc 415	. . . . . . . . . . . . . . 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 943	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁))
81	80	intnand 943	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁)))
82		elfz2 10397	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ (𝑀 ≤ 𝑝 ∧ 𝑝 ≤ 𝑁)))
83	81, 82	sylnibr 688	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ∈ (𝑀...𝑁))
84	65, 83	ssneldd 3251	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ¬ 𝑝 ∈ 𝐴)
85	84	iffalsed 3647	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) = 1)
86	85, 44	eqeltrdi 2329	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ)
87		nfcv 2392	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑝
88		nfv 1581	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑝 ∈ 𝐴
89		nfcsb1v 3180	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵
90	88, 89, 49	nfif 3666	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)
91		eleq1w 2299	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → (𝑘 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴))
92		csbeq1a 3156	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → 𝐵 = ⦋𝑝 / 𝑘⦌𝐵)
93	91, 92	ifbieq1d 3660	. . . . . . . . 9 ⊢ (𝑘 = 𝑝 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
94	87, 90, 93, 54	fvmptf 5792	. . . . . . . 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 415	. . . . . . 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 2315	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → ((𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) ∈ ℂ)
97	57	fvconst2 5922	. . . . . . . 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 2329	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑝 ∈ (ℤ_≥‘(𝑁 + 1))) → (((ℤ_≥‘(𝑁 + 1)) × {1})‘𝑝) ∈ ℂ)
100		mulcl 8296	. . . . . . 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 10894	. . . . 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 12287	. . . . . 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 2271	. . . 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 12034	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 1)
107		breq1 4128	. . . . 5 ⊢ (𝑦 = 1 → (𝑦 # 0 ↔ 1 # 0))
108		breq2 4129	. . . . 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 477	. . . 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 2920	. . 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 418	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq(𝑁 + 1)( · , (𝑘 ∈ 𝑍 ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
112		seqeq1 10865	. . . . . 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 4135	. . . . 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 468	. . . 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 1878	. . 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 2929	. 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 415	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 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ∃wrex 2529 Vcvv 2821 ⦋csb 3147 ⊆ wss 3220 ifcif 3635 {csn 3705 class class class wbr 4125 ↦ cmpt 4187 × cxp 4767 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 ℝcr 8168 0cc0 8169 1c1 8170 + caddc 8172 · cmul 8174 < clt 8350 ≤ cle 8351 # cap 8899 ℤcz 9623 ℤ_≥cuz 9900 ...cfz 10390 seqcseq 10862 ⇝ cli 12022

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-n0 9543 df-z 9624 df-uz 9901 df-rp 10034 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-cj 11585 df-rsqrt 11742 df-abs 11743 df-clim 12023

This theorem is referenced by: fprodssdc 12335

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