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Theorem infpnlem1 12497

Description: Lemma for infpn 12499. The smallest divisor (greater than 1) 𝑀 of 𝑁! + 1 is a prime greater than 𝑁. (Contributed by NM, 5-May-2005.)

**Hypothesis**
Ref	Expression
infpnlem.1	⊢ 𝐾 = ((!‘𝑁) + 1)

**Assertion**
Ref	Expression
infpnlem1	⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗)) → (𝑁 < 𝑀 ∧ ∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀)))))

Distinct variable groups: 𝑗,𝑁 𝑗,𝑀 𝑗,𝐾

**Proof of Theorem infpnlem1**
Step	Hyp	Ref	Expression
1		nnz 9336	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → 𝑁 ∈ ℤ)
3		nnz 9336	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
4	3	ad2antlr 489	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → 𝑀 ∈ ℤ)
5		zdclt 9394	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
6	2, 4, 5	syl2anc 411	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → DECID 𝑁 < 𝑀)
7		nnre 8989	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
8		nnre 8989	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
9		lenlt 8095	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
10	7, 8, 9	syl2anr 290	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
11	10	adantr 276	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
12		nnnn0 9247	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
13		facndiv 10810	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
14		infpnlem.1	. . . . . . . . . . 11 ⊢ 𝐾 = ((!‘𝑁) + 1)
15	14	oveq1i 5928	. . . . . . . . . 10 ⊢ (𝐾 / 𝑀) = (((!‘𝑁) + 1) / 𝑀)
16		nnz 9336	. . . . . . . . . 10 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (𝐾 / 𝑀) ∈ ℤ)
17	15, 16	eqeltrrid 2281	. . . . . . . . 9 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
18	13, 17	nsyl 629	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (𝐾 / 𝑀) ∈ ℕ)
19	12, 18	sylanl1 402	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (𝐾 / 𝑀) ∈ ℕ)
20	19	expr 375	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → (𝑀 ≤ 𝑁 → ¬ (𝐾 / 𝑀) ∈ ℕ))
21	11, 20	sylbird 170	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → (¬ 𝑁 < 𝑀 → ¬ (𝐾 / 𝑀) ∈ ℕ))
22		condc 854	. . . . 5 ⊢ (DECID 𝑁 < 𝑀 → ((¬ 𝑁 < 𝑀 → ¬ (𝐾 / 𝑀) ∈ ℕ) → ((𝐾 / 𝑀) ∈ ℕ → 𝑁 < 𝑀)))
23	6, 21, 22	sylc 62	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → ((𝐾 / 𝑀) ∈ ℕ → 𝑁 < 𝑀))
24	23	expimpd 363	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) → 𝑁 < 𝑀))
25	24	adantrd 279	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗)) → 𝑁 < 𝑀))
26	12	faccld 10807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ)
27	26	peano2nnd 8997	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ)
28	14, 27	eqeltrid 2280	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ)
29	28	nncnd 8996	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℂ)
30		nndivtr 9024	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝐾 ∈ ℂ) ∧ ((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ)) → (𝐾 / 𝑗) ∈ ℕ)
31	30	ex 115	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝐾 ∈ ℂ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
32	31	3com13 1210	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
33	32	3expa 1205	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
34	29, 33	sylanl1 402	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
35	34	adantrl 478	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
36		nnre 8989	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
37		letri3 8100	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑗 = 𝑀 ↔ (𝑗 ≤ 𝑀 ∧ 𝑀 ≤ 𝑗)))
38	36, 7, 37	syl2an 289	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑗 = 𝑀 ↔ (𝑗 ≤ 𝑀 ∧ 𝑀 ≤ 𝑗)))
39	38	biimprd 158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((𝑗 ≤ 𝑀 ∧ 𝑀 ≤ 𝑗) → 𝑗 = 𝑀))
40	39	exp4b 367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ → (𝑀 ∈ ℕ → (𝑗 ≤ 𝑀 → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))))
41	40	com3l 81	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ → (𝑗 ≤ 𝑀 → (𝑗 ∈ ℕ → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))))
42	41	imp32 257	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))
43	42	adantll 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))
44	43	imim2d 54	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
45	44	com23 78	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))
46	35, 45	sylan2d 294	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → ((1 < 𝑗 ∧ ((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ)) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))
47	46	exp4d 369	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (1 < 𝑗 → ((𝑀 / 𝑗) ∈ ℕ → ((𝐾 / 𝑀) ∈ ℕ → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))))
48	47	com24 87	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → ((𝐾 / 𝑀) ∈ ℕ → ((𝑀 / 𝑗) ∈ ℕ → (1 < 𝑗 → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))))
49	48	exp32 365	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑗 ≤ 𝑀 → (𝑗 ∈ ℕ → ((𝐾 / 𝑀) ∈ ℕ → ((𝑀 / 𝑗) ∈ ℕ → (1 < 𝑗 → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))))))
50	49	com24 87	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((𝐾 / 𝑀) ∈ ℕ → (𝑗 ∈ ℕ → (𝑗 ≤ 𝑀 → ((𝑀 / 𝑗) ∈ ℕ → (1 < 𝑗 → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))))))
51	50	imp31 256	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝑗 ≤ 𝑀 → ((𝑀 / 𝑗) ∈ ℕ → (1 < 𝑗 → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))))
52	51	com14 88	. . . . . . . . 9 ⊢ (1 < 𝑗 → (𝑗 ≤ 𝑀 → ((𝑀 / 𝑗) ∈ ℕ → ((((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))))
53	52	3imp 1195	. . . . . . . 8 ⊢ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → ((((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))
54	53	com3l 81	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
55	54	ralimdva 2561	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝐾 / 𝑀) ∈ ℕ) → (∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
56	55	ex 115	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((𝐾 / 𝑀) ∈ ℕ → (∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀))))
57	56	adantld 278	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) → (∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀))))
58	57	impd 254	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗)) → ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
59		prime 9416	. . . 4 ⊢ (𝑀 ∈ ℕ → (∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀)) ↔ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
60	59	adantl 277	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀)) ↔ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
61	58, 60	sylibrd 169	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗)) → ∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀))))
62	25, 61	jcad 307	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗)) → (𝑁 < 𝑀 ∧ ∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀)))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ∀wral 2472 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 ℝcr 7871 1c1 7873 + caddc 7875 < clt 8054 ≤ cle 8055 / cdiv 8691 ℕcn 8982 ℕ₀cn0 9240 ℤcz 9317 !cfa 10796

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-seqfrec 10519 df-fac 10797

This theorem is referenced by: infpnlem2 12498

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