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Theorem infpnlem1 13116

Description: Lemma for infpn 13118. 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 9642	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
2	1	ad2antrr 492	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → 𝑁 ∈ ℤ)
3		nnz 9642	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
4	3	ad2antlr 493	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → 𝑀 ∈ ℤ)
5		zdclt 9701	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
6	2, 4, 5	syl2anc 415	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → DECID 𝑁 < 𝑀)
7		nnre 9290	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
8		nnre 9290	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
9		lenlt 8391	. . . . . . . 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 9549	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
13		facndiv 11155	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
14		infpnlem.1	. . . . . . . . . . 11 ⊢ 𝐾 = ((!‘𝑁) + 1)
15	14	oveq1i 6085	. . . . . . . . . 10 ⊢ (𝐾 / 𝑀) = (((!‘𝑁) + 1) / 𝑀)
16		nnz 9642	. . . . . . . . . 10 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (𝐾 / 𝑀) ∈ ℤ)
17	15, 16	eqeltrrid 2326	. . . . . . . . 9 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
18	13, 17	nsyl 637	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (𝐾 / 𝑀) ∈ ℕ)
19	12, 18	sylanl1 406	. . . . . . 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 865	. . . . 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 11152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ)
27	26	peano2nnd 9298	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ)
28	14, 27	eqeltrid 2325	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ)
29	28	nncnd 9297	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℂ)
30		nndivtr 9325	. . . . . . . . . . . . . . . . . . . . 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 1239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
33	32	3expa 1234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
34	29, 33	sylanl1 406	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
35	34	adantrl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
36		nnre 9290	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
37		letri3 8396	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 1224	. . . . . . . 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 2617	. . . . . 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 9724	. . . 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 720 DECID wdc 846 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 ℝcr 8168 1c1 8170 + caddc 8172 < clt 8350 ≤ cle 8351 / cdiv 8992 ℕcn 9283 ℕ₀cn0 9542 ℤcz 9623 !cfa 11141

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-stab 843 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-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-n0 9543 df-z 9624 df-uz 9901 df-seqfrec 10863 df-fac 11142

This theorem is referenced by: infpnlem2 13117

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