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Theorem prmind2 11837

Description: A variation on prmind 11838 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 5790	. . . 4 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
3	2	raleqdv 2635	. . 3 ⊢ (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑))
4		oveq2 5790	. . . 4 ⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
5	4	raleqdv 2635	. . 3 ⊢ (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑))
6		oveq2 5790	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
7	6	raleqdv 2635	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
8		oveq2 5790	. . . 4 ⊢ (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴))
9	8	raleqdv 2635	. . 3 ⊢ (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑))
10		prmind.6	. . . . 5 ⊢ 𝜓
11		elfz1eq 9846	. . . . . 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 2488	. . 3 ⊢ ∀𝑥 ∈ (1...1)𝜑
16		peano2nn 8756	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
17	16	ad2antrr 480	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ)
18	17	nncnd 8758	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ)
19		elfzuz 9833	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ (ℤ_≥‘2))
20	19	ad2antrl 482	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (ℤ_≥‘2))
21		eluz2nn 9388	. . . . . . . . . . . . 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 8758	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ)
24	22	nnap0d 8790	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 # 0)
25	18, 23, 24	divcanap2d 8576	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1))
26		simprr 522	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1))
27	22	nnzd 9196	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ)
28	22	nnne0d 8789	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0)
29	17	nnzd 9196	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
30		dvdsval2 11532	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
31	27, 28, 29, 30	syl3anc 1217	. . . . . . . . . . . . 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 7808	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦)
34		elfzle2 9839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1))
35	34	ad2antrl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1))
36		nncn 8752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
37	36	ad2antrr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ)
38		ax-1cn 7737	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
39		pncan 7992	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
40	37, 38, 39	sylancl 410	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘)
41	35, 40	breqtrd 3962	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ 𝑘)
42		nnz 9097	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
43	42	ad2antrr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ)
44		zleltp1 9133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
45	27, 43, 44	syl2anc 409	. . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1))
48		1red 7805	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈ ℝ)
49	17	nnred 8757	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
50	22	nnred 8757	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ)
51	22	nngt0d 8788	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦)
52		ltmuldiv 8656	. . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . 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 9422	. . . . . . . . . . . 12 ⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2) ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 < ((𝑘 + 1) / 𝑦)))
56	32, 54, 55	sylanbrc 414	. . . . . . . . . . 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 520	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑)
59		fznn 9900	. . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘))
62	57, 58, 61	rspcdva 2798	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒)
63		vex 2692	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
64		prmind.3	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))
65	63, 64	sbcie 2947	. . . . . . . . . . . . . 14 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝜃)
66		dfsbcq 2915	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
67	65, 66	bitr3id 193	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
68	64	cbvralv 2657	. . . . . . . . . . . . . 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 9511	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ⁺)
71	22	nnrpd 9511	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ⁺)
72	70, 71	rpdivcld 9531	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℝ⁺)
73	72	rpgt0d 9516	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦))
74		elnnz 9088	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦)))
75	32, 73, 74	sylanbrc 414	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ)
76	17	nnap0d 8790	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) # 0)
77	18, 76	dividapd 8570	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1)
78		eluz2gt1 9423	. . . . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦)
81	17	nngt0d 8788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1))
82		ltdiv23 8674	. . . . . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . . . 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 9133	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
86	32, 43, 85	syl2anc 409	. . . . . . . . . . . . . . 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 9900	. . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)
92	62, 91	jca 304	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
93	67	anbi2d 460	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)))
94		oveq2 5790	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦)))
95	94	sbceq1d 2918	. . . . . . . . . . . . . 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 9359	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 𝑦 ∈ ℤ)
101	100	adantl 275	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → 𝑦 ∈ ℤ)
102		eluzelz 9359	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
103	102	adantr 274	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℤ)
104	101, 103	zmulcld 9203	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑦 ∈ (ℤ_≥‘2)) → (𝑦 · 𝑧) ∈ ℤ)
105		prmind.4	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏))
106	105	sbcieg 2945	. . . . . . . . . . . . . . 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 2755	. . . . . . . . . . 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 2919	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑)
113	112	rexlimdvaa 2553	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
114		ralnex 2427	. . . . . . . . 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 9386	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
117	115, 116	sylib 121	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ (ℤ_≥‘1))
118		eluzp1p1 9375	. . . . . . . . . . . . 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 8803	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
121	120	fveq2i 5432	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
122	119, 121	eleqtrrdi 2234	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ_≥‘2))
123		isprm3 11835	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ ℙ ↔ ((𝑘 + 1) ∈ (ℤ_≥‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1)))
124	123	baibr 906	. . . . . . . . . . 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 2657	. . . . . . . . . . . . 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 8758	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ)
130	129, 38, 39	sylancl 410	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
131	130	oveq2d 5798	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘))
132	131	raleqdv 2635	. . . . . . . . . . . 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 2282	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 + 1)
135		nfv 1509	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒
136		nfsbc1v 2931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
137	135, 136	nfim 1552	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)
138		oveq1 5789	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
139	138	oveq2d 5798	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1)))
140	139	raleqdv 2635	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒))
141		sbceq1a 2922	. . . . . . . . . . . . 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 2754	. . . . . . . . . . 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 9106	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
150	149	a1i 9	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 2 ∈ ℤ)
151	115	nnzd 9196	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℤ)
152	151	peano2zd 9200	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
153		1zzd 9105	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 1 ∈ ℤ)
154	152, 153	zsubcld 9202	. . . . . . . . . 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 11537	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑘 + 1) ∈ ℤ) → DECID 𝑦 ∥ (𝑘 + 1))
157	155, 152, 156	syl2anr 288	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ 𝑦 ∈ (2...((𝑘 + 1) − 1))) → DECID 𝑦 ∥ (𝑘 + 1))
158	150, 154, 157	exfzdc 10048	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → DECID ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
159		exmiddc 822	. . . . . . . . 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 708	. . . . . . 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 3568	. . . . . . 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 323	. . . 4 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
167		fzsuc 9880	. . . . . . 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 2635	. . . . 5 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑))
170		ralunb 3262	. . . . 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 8760	. 2 ⊢ (𝐴 ∈ ℕ → ∀𝑥 ∈ (1...𝐴)𝜑)
174		elfz1end 9866	. . 3 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
175	174	biimpi 119	. 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴))
176	1, 173, 175	rspcdva 2798	1 ⊢ (𝐴 ∈ ℕ → 𝜂)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 820 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ∀wral 2417 ∃wrex 2418 [wsbc 2913 ∪ cun 3074 {csn 3532 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 < clt 7824 ≤ cle 7825 − cmin 7957 / cdiv 8456 ℕcn 8744 2c2 8795 ℤcz 9078 ℤ_≥cuz 9350 ...cfz 9821 ∥ cdvds 11529 ℙcprime 11824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-1o 6321 df-2o 6322 df-er 6437 df-en 6643 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-fzo 9951 df-fl 10074 df-mod 10127 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-dvds 11530 df-prm 11825

This theorem is referenced by: prmind 11838

W3C validator