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Theorem prmind2 16622

Description: A variation on prmind 16623 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 7417	. . . 4 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
3	2	raleqdv 3326	. . 3 ⊢ (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑))
4		oveq2 7417	. . . 4 ⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
5	4	raleqdv 3326	. . 3 ⊢ (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑))
6		oveq2 7417	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
7	6	raleqdv 3326	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
8		oveq2 7417	. . . 4 ⊢ (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴))
9	8	raleqdv 3326	. . 3 ⊢ (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑))
10		prmind.6	. . . . 5 ⊢ 𝜓
11		elfz1eq 13512	. . . . . 6 ⊢ (𝑥 ∈ (1...1) → 𝑥 = 1)
12		prmind.1	. . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
13	11, 12	syl 17	. . . . 5 ⊢ (𝑥 ∈ (1...1) → (𝜑 ↔ 𝜓))
14	10, 13	mpbiri 258	. . . 4 ⊢ (𝑥 ∈ (1...1) → 𝜑)
15	14	rgen 3064	. . 3 ⊢ ∀𝑥 ∈ (1...1)𝜑
16		peano2nn 12224	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
17	16	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ)
18	17	nncnd 12228	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ)
19		elfzuz 13497	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ (ℤ_≥‘2))
20	19	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (ℤ_≥‘2))
21		eluz2nn 12868	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 𝑦 ∈ ℕ)
22	20, 21	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℕ)
23	22	nncnd 12228	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ)
24	22	nnne0d 12262	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0)
25	18, 23, 24	divcan2d 11992	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1))
26		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1))
27	22	nnzd 12585	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ)
28	17	nnzd 12585	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
29		dvdsval2 16200	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
30	27, 24, 28, 29	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
31	26, 30	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ)
32	23	mullidd 11232	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦)
33		elfzle2 13505	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1))
34	33	ad2antrl 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1))
35		nncn 12220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
36	35	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ)
37		ax-1cn 11168	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
38		pncan 11466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
39	36, 37, 38	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘)
40	34, 39	breqtrd 5175	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ 𝑘)
41		nnz 12579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
42	41	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ)
43		zleltp1 12613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
44	27, 42, 43	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
45	40, 44	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1))
46	32, 45	eqbrtrd 5171	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1))
47		1red 11215	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈ ℝ)
48	17	nnred 12227	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
49	22	nnred 12227	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ)
50	22	nngt0d 12261	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦)
51		ltmuldiv 12087	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
52	47, 48, 49, 50, 51	syl112anc 1375	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
53	46, 52	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦))
54		eluz2b1 12903	. . . . . . . . . . . 12 ⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2) ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 < ((𝑘 + 1) / 𝑦)))
55	31, 53, 54	sylanbrc 584	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2))
56		prmind.2	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
57		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑)
58		fznn 13569	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘)))
59	42, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘)))
60	22, 40, 59	mpbir2and 712	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘))
61	56, 57, 60	rspcdva 3614	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒)
62		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
63		prmind.3	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))
64	62, 63	sbcie 3821	. . . . . . . . . . . . . 14 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝜃)
65		dfsbcq 3780	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
66	64, 65	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
67	63	cbvralvw 3235	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃)
68	57, 67	sylib 217	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃)
69	17	nnrpd 13014	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ⁺)
70	22	nnrpd 13014	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ⁺)
71	69, 70	rpdivcld 13033	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℝ⁺)
72	71	rpgt0d 13019	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦))
73		elnnz 12568	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦)))
74	31, 72, 73	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ)
75	17	nnne0d 12262	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ≠ 0)
76	18, 75	dividd 11988	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1)
77		eluz2gt1 12904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 1 < 𝑦)
78	20, 77	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦)
79	76, 78	eqbrtrd 5171	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦)
80	17	nngt0d 12261	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1))
81		ltdiv23 12105	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 + 1) ∈ ℝ ∧ ((𝑘 + 1) ∈ ℝ ∧ 0 < (𝑘 + 1)) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
82	48, 48, 80, 49, 50, 81	syl122anc 1380	. . . . . . . . . . . . . . . 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)))
83	79, 82	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1))
84		zleltp1 12613	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
85	31, 42, 84	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
86	83, 85	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘)
87		fznn 13569	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘)))
88	42, 87	syl 17	. . . . . . . . . . . . . 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)))
89	74, 86, 88	mpbir2and 712	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘))
90	66, 68, 89	rspcdva 3614	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)
91	61, 90	jca 513	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
92	66	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)))
93		ovex 7442	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 · 𝑧) ∈ V
94		prmind.4	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏))
95	93, 94	sbcie 3821	. . . . . . . . . . . . . . 15 ⊢ ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ 𝜏)
96		oveq2 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦)))
97	96	sbceq1d 3783	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
98	95, 97	bitr3id 285	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜏 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
99	92, 98	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒 ∧ 𝜃) → 𝜏) ↔ ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
100	99	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ 𝜃) → 𝜏)) ↔ (𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))))
101		prmind2.8	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏))
102	101	expcom 415	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℤ_≥‘2) → (𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ 𝜃) → 𝜏)))
103	100, 102	vtoclga 3566	. . . . . . . . . . 11 ⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ_≥‘2) → (𝑦 ∈ (ℤ_≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
104	55, 20, 91, 103	syl3c 66	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)
105	25, 104	sbceq1dd 3784	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑)
106	105	rexlimdvaa 3157	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
107		ralnex 3073	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
108		simpl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ)
109		elnnuz 12866	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
110	108, 109	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ (ℤ_≥‘1))
111		eluzp1p1 12850	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (𝑘 + 1) ∈ (ℤ_≥‘(1 + 1)))
112	110, 111	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ_≥‘(1 + 1)))
113		df-2 12275	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
114	113	fveq2i 6895	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
115	112, 114	eleqtrrdi 2845	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ_≥‘2))
116		isprm3 16620	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ ℙ ↔ ((𝑘 + 1) ∈ (ℤ_≥‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1)))
117	116	baibr 538	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
118	115, 117	syl 17	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
119		simpr 486	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑)
120	56	cbvralvw 3235	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)
121	119, 120	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒)
122	108	nncnd 12228	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ)
123	122, 37, 38	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
124	123	oveq2d 7425	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘))
125	124	raleqdv 3326	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒))
126	121, 125	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)
127		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 + 1)
128		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒
129		nfsbc1v 3798	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
130	128, 129	nfim 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)
131		oveq1 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
132	131	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1)))
133	132	raleqdv 3326	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒))
134		sbceq1a 3789	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
135	133, 134	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)))
136		prmind2.7	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)
137	136	ex 414	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℙ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑))
138	127, 130, 135, 137	vtoclgaf 3565	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ ℙ → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑))
139	126, 138	syl5com 31	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑))
140	118, 139	sylbid 239	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
141	107, 140	biimtrrid 242	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
142	106, 141	pm2.61d 179	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
143	142	ex 414	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → [(𝑘 + 1) / 𝑥]𝜑))
144		ralsnsg 4673	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
145	16, 144	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
146	143, 145	sylibrd 259	. . . . 5 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
147	146	ancld 552	. . . 4 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
148		fzsuc 13548	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
149	109, 148	sylbi 216	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
150	149	raleqdv 3326	. . . . 5 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑))
151		ralunb 4192	. . . . 5 ⊢ (∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
152	150, 151	bitrdi 287	. . . 4 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
153	147, 152	sylibrd 259	. . 3 ⊢ (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
154	3, 5, 7, 9, 15, 153	nnind 12230	. 2 ⊢ (𝐴 ∈ ℕ → ∀𝑥 ∈ (1...𝐴)𝜑)
155		elfz1end 13531	. . 3 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
156	155	biimpi 215	. 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴))
157	1, 154, 156	rspcdva 3614	1 ⊢ (𝐴 ∈ ℕ → 𝜂)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 [wsbc 3778 ∪ cun 3947 {csn 4629 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 2c2 12267 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 ∥ cdvds 16197 ℙcprime 16608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-prm 16609

This theorem is referenced by: prmind 16623 4sqlem19 16896

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