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Theorem prmind2 11594

Description: A variation on prmind 11595 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)

**Hypotheses**
Ref	Expression
prmind.1	⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
prmind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
prmind.3	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))
prmind.4	⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏))
prmind.5	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂))
prmind.6	⊢ 𝜓
prmind2.7	⊢ ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)
prmind2.8	⊢ ((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏))

**Assertion**
Ref	Expression
prmind2	⊢ (𝐴 ∈ ℕ → 𝜂)

Distinct variable groups: 𝑥,𝑦 𝑥,𝐴 𝑥,𝑧,𝜒 𝜂,𝑥 𝜏,𝑥 𝜃,𝑥 𝑦,𝑧,𝜑 Allowed substitution hints: 𝜑(𝑥) 𝜓(𝑥,𝑦,𝑧) 𝜒(𝑦) 𝜃(𝑦,𝑧) 𝜏(𝑦,𝑧) 𝜂(𝑦,𝑧) 𝐴(𝑦,𝑧)

Proof of Theorem prmind2 Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmind.5	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂))
2		oveq2 5714	. . . 4 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
3	2	raleqdv 2590	. . 3 ⊢ (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑))
4		oveq2 5714	. . . 4 ⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
5	4	raleqdv 2590	. . 3 ⊢ (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑))
6		oveq2 5714	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
7	6	raleqdv 2590	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
8		oveq2 5714	. . . 4 ⊢ (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴))
9	8	raleqdv 2590	. . 3 ⊢ (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑))
10		prmind.6	. . . . 5 ⊢ 𝜓
11		elfz1eq 9656	. . . . . 6 ⊢ (𝑥 ∈ (1...1) → 𝑥 = 1)
12		prmind.1	. . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
13	11, 12	syl 14	. . . . 5 ⊢ (𝑥 ∈ (1...1) → (𝜑 ↔ 𝜓))
14	10, 13	mpbiri 167	. . . 4 ⊢ (𝑥 ∈ (1...1) → 𝜑)
15	14	rgen 2444	. . 3 ⊢ ∀𝑥 ∈ (1...1)𝜑
16		peano2nn 8590	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
17	16	ad2antrr 475	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ)
18	17	nncnd 8592	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ)
19		elfzuz 9643	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ (ℤ_≥‘2))
20	19	ad2antrl 477	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (ℤ_≥‘2))
21		eluz2nn 9214	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 𝑦 ∈ ℕ)
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℕ)
23	22	nncnd 8592	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ)
24	22	nnap0d 8624	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 # 0)
25	18, 23, 24	divcanap2d 8413	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1))
26		simprr 502	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1))
27	22	nnzd 9024	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ)
28	22	nnne0d 8623	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0)
29	17	nnzd 9024	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
30		dvdsval2 11291	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
31	27, 28, 29, 30	syl3anc 1184	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
32	26, 31	mpbid 146	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ)
33	23	mulid2d 7656	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦)
34		elfzle2 9649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1))
35	34	ad2antrl 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1))
36		nncn 8586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
37	36	ad2antrr 475	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ)
38		ax-1cn 7588	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
39		pncan 7839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
40	37, 38, 39	sylancl 407	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘)
41	35, 40	breqtrd 3899	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ 𝑘)
42		nnz 8925	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
43	42	ad2antrr 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ)
44		zleltp1 8961	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
45	27, 43, 44	syl2anc 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
46	41, 45	mpbid 146	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1))
47	33, 46	eqbrtrd 3895	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1))
48		1red 7653	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈ ℝ)
49	17	nnred 8591	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
50	22	nnred 8591	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ)
51	22	nngt0d 8622	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦)
52		ltmuldiv 8490	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
53	48, 49, 50, 51, 52	syl112anc 1188	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
54	47, 53	mpbid 146	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦))
55		eluz2b1 9245	. . . . . . . . . . . 12 ⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2) ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 < ((𝑘 + 1) / 𝑦)))
56	32, 54, 55	sylanbrc 411	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2))
57		prmind.2	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
58		simplr 500	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑)
59		fznn 9710	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘)))
60	43, 59	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘)))
61	22, 41, 60	mpbir2and 896	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘))
62	57, 58, 61	rspcdva 2749	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒)
63		vex 2644	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
64		prmind.3	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))
65	63, 64	sbcie 2895	. . . . . . . . . . . . . 14 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝜃)
66		dfsbcq 2864	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
67	65, 66	syl5bbr 193	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
68	64	cbvralv 2612	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃)
69	58, 68	sylib 121	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃)
70	17	nnrpd 9329	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ⁺)
71	22	nnrpd 9329	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ⁺)
72	70, 71	rpdivcld 9348	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℝ⁺)
73	72	rpgt0d 9333	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦))
74		elnnz 8916	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦)))
75	32, 73, 74	sylanbrc 411	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ)
76	17	nnap0d 8624	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) # 0)
77	18, 76	dividapd 8407	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1)
78		eluz2gt1 9246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 1 < 𝑦)
79	20, 78	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦)
80	77, 79	eqbrtrd 3895	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦)
81	17	nngt0d 8622	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1))
82		ltdiv23 8508	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 + 1) ∈ ℝ ∧ ((𝑘 + 1) ∈ ℝ ∧ 0 < (𝑘 + 1)) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
83	49, 49, 81, 50, 51, 82	syl122anc 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
84	80, 83	mpbid 146	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1))
85		zleltp1 8961	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
86	32, 43, 85	syl2anc 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
87	84, 86	mpbird 166	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘)
88		fznn 9710	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘)))
89	43, 88	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘)))
90	75, 87, 89	mpbir2and 896	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘))
91	67, 69, 90	rspcdva 2749	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)
92	62, 91	jca 302	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
93	67	anbi2d 455	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)))
94		oveq2 5714	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦)))
95	94	sbceq1d 2867	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
96	93, 95	imbi12d 233	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒 ∧ 𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑) ↔ ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
97	96	imbi2d 229	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ 𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑)) ↔ (𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))))
98		prmind2.8	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏))
99	98	ancoms 266	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏))
100		eluzelz 9185	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 𝑦 ∈ ℤ)
101	100	adantl 273	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → 𝑦 ∈ ℤ)
102		eluzelz 9185	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
103	102	adantr 272	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℤ)
104	101, 103	zmulcld 9031	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → (𝑦 · 𝑧) ∈ ℤ)
105		prmind.4	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏))
106	105	sbcieg 2893	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 · 𝑧) ∈ ℤ → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ 𝜏))
107	104, 106	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ 𝜏))
108	99, 107	sylibrd 168	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → ((𝜒 ∧ 𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑))
109	108	ex 114	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℤ_≥‘2) → (𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ 𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑)))
110	97, 109	vtoclga 2707	. . . . . . . . . . 11 ⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2) → (𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
111	56, 20, 92, 110	syl3c 63	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)
112	25, 111	sbceq1dd 2868	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑)
113	112	rexlimdvaa 2509	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
114		ralnex 2385	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
115		simpl 108	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ)
116		elnnuz 9212	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
117	115, 116	sylib 121	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ (ℤ_≥‘1))
118		eluzp1p1 9201	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (𝑘 + 1) ∈ (ℤ_≥‘(1 + 1)))
119	117, 118	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ_≥‘(1 + 1)))
120		df-2 8637	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
121	120	fveq2i 5356	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
122	119, 121	syl6eleqr 2193	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ_≥‘2))
123		isprm3 11592	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ ℙ ↔ ((𝑘 + 1) ∈ (ℤ_≥‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1)))
124	123	baibr 873	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
125	122, 124	syl 14	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
126		simpr 109	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑)
127	57	cbvralv 2612	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)
128	126, 127	sylib 121	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒)
129	115	nncnd 8592	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ)
130	129, 38, 39	sylancl 407	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
131	130	oveq2d 5722	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘))
132	131	raleqdv 2590	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒))
133	128, 132	mpbird 166	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)
134		nfcv 2240	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 + 1)
135		nfv 1476	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒
136		nfsbc1v 2880	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
137	135, 136	nfim 1519	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)
138		oveq1 5713	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
139	138	oveq2d 5722	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1)))
140	139	raleqdv 2590	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒))
141		sbceq1a 2871	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
142	140, 141	imbi12d 233	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)))
143		prmind2.7	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)
144	143	ex 114	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℙ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑))
145	134, 137, 142, 144	vtoclgaf 2706	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ ℙ → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑))
146	133, 145	syl5com 29	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑))
147	125, 146	sylbid 149	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
148	114, 147	syl5bir 152	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
149		2z 8934	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
150	149	a1i 9	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 2 ∈ ℤ)
151	115	nnzd 9024	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℤ)
152	151	peano2zd 9028	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
153		1zzd 8933	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 1 ∈ ℤ)
154	152, 153	zsubcld 9030	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) ∈ ℤ)
155	19, 21	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ ℕ)
156		dvdsdc 11296	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑘 + 1) ∈ ℤ) → DECID 𝑦 ∥ (𝑘 + 1))
157	155, 152, 156	syl2anr 286	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ 𝑦 ∈ (2...((𝑘 + 1) − 1))) → DECID 𝑦 ∥ (𝑘 + 1))
158	150, 154, 157	exfzdc 9858	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → DECID ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
159		exmiddc 788	. . . . . . . . 9 ⊢ (DECID ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) ∨ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)))
160	158, 159	syl 14	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) ∨ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)))
161	113, 148, 160	mpjaod 679	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
162	161	ex 114	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → [(𝑘 + 1) / 𝑥]𝜑))
163		ralsnsg 3508	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
164	16, 163	syl 14	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
165	162, 164	sylibrd 168	. . . . 5 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
166	165	ancld 321	. . . 4 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
167		fzsuc 9690	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
168	116, 167	sylbi 120	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
169	168	raleqdv 2590	. . . . 5 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑))
170		ralunb 3204	. . . . 5 ⊢ (∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
171	169, 170	syl6bb 195	. . . 4 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
172	166, 171	sylibrd 168	. . 3 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
173	3, 5, 7, 9, 15, 172	nnind 8594	. 2 ⊢ (𝐴 ∈ ℕ → ∀𝑥 ∈ (1...𝐴)𝜑)
174		elfz1end 9676	. . 3 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
175	174	biimpi 119	. 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴))
176	1, 173, 175	rspcdva 2749	1 ⊢ (𝐴 ∈ ℕ → 𝜂)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 670 DECID wdc 786 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 ∀wral 2375 ∃wrex 2376 [wsbc 2862 ∪ cun 3019 {csn 3474 class class class wbr 3875 ‘cfv 5059 (class class class)co 5706 ℂcc 7498 ℝcr 7499 0cc0 7500 1c1 7501 + caddc 7503 · cmul 7505 < clt 7672 ≤ cle 7673 − cmin 7804 / cdiv 8293 ℕcn 8578 2c2 8629 ℤcz 8906 ℤ_≥cuz 9176 ...cfz 9631 ∥ cdvds 11288 ℙcprime 11581

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-1o 6243 df-2o 6244 df-er 6359 df-en 6565 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-fz 9632 df-fzo 9761 df-fl 9884 df-mod 9937 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-dvds 11289 df-prm 11582

This theorem is referenced by: prmind 11595

W3C validator