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Theorem infpnlem1 15985

Description: Lemma for infpn 15987. 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		nnre 11358	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
2		nnre 11358	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
3		lenlt 10435	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
4	1, 2, 3	syl2anr 592	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
5	4	adantr 474	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
6		nnnn0 11626	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		facndiv 13368	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
8		infpnlem.1	. . . . . . . . . . 11 ⊢ 𝐾 = ((!‘𝑁) + 1)
9	8	oveq1i 6915	. . . . . . . . . 10 ⊢ (𝐾 / 𝑀) = (((!‘𝑁) + 1) / 𝑀)
10		nnz 11727	. . . . . . . . . 10 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (𝐾 / 𝑀) ∈ ℤ)
11	9, 10	syl5eqelr 2911	. . . . . . . . 9 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
12	7, 11	nsyl 138	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (𝐾 / 𝑀) ∈ ℕ)
13	6, 12	sylanl1 672	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (𝐾 / 𝑀) ∈ ℕ)
14	13	expr 450	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → (𝑀 ≤ 𝑁 → ¬ (𝐾 / 𝑀) ∈ ℕ))
15	5, 14	sylbird 252	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → (¬ 𝑁 < 𝑀 → ¬ (𝐾 / 𝑀) ∈ ℕ))
16	15	con4d 115	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → ((𝐾 / 𝑀) ∈ ℕ → 𝑁 < 𝑀))
17	16	expimpd 447	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) → 𝑁 < 𝑀))
18	17	adantrd 487	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗)) → 𝑁 < 𝑀))
19		faccl 13363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ)
20	6, 19	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ)
21	20	peano2nnd 11369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ)
22	8, 21	syl5eqel 2910	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ)
23	22	nncnd 11368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℂ)
24		nndivtr 11398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝐾 ∈ ℂ) ∧ ((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ)) → (𝐾 / 𝑗) ∈ ℕ)
25	24	ex 403	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝐾 ∈ ℂ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
26	25	3com13 1160	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
27	26	3expa 1153	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
28	23, 27	sylanl1 672	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
29	28	adantrl 709	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
30		nnre 11358	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
31		letri3 10442	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑗 = 𝑀 ↔ (𝑗 ≤ 𝑀 ∧ 𝑀 ≤ 𝑗)))
32	30, 1, 31	syl2an 591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑗 = 𝑀 ↔ (𝑗 ≤ 𝑀 ∧ 𝑀 ≤ 𝑗)))
33	32	biimprd 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((𝑗 ≤ 𝑀 ∧ 𝑀 ≤ 𝑗) → 𝑗 = 𝑀))
34	33	exp4b 423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ → (𝑀 ∈ ℕ → (𝑗 ≤ 𝑀 → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))))
35	34	com3l 89	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ → (𝑗 ≤ 𝑀 → (𝑗 ∈ ℕ → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))))
36	35	imp32 411	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))
37	36	adantll 707	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (𝑀 ≤ 𝑗 → 𝑗 = 𝑀))
38	37	imim2d 57	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
39	38	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ (𝑗 ≤ 𝑀 ∧ 𝑗 ∈ ℕ)) → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀 ≤ 𝑗) → 𝑗 = 𝑀)))
40	29, 39	sylan2d 600	. . . . . . . . . . . . . . 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)))
41	40	exp4d 426	. . . . . . . . . . . . . 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)))))
42	41	com24 95	. . . . . . . . . . . . 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)))))
43	42	exp32 413	. . . . . . . . . . . 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)))))))
44	43	com24 95	. . . . . . . . . . 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)))))))
45	44	imp31 410	. . . . . . . . . 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)))))
46	45	com14 96	. . . . . . . . 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)))))
47	46	3imp 1143	. . . . . . . 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)))
48	47	com3l 89	. . . . . . 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)))
49	48	ralimdva 3171	. . . . . 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)))
50	49	ex 403	. . . . 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))))
51	50	adantld 486	. . . 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))))
52	51	impd 400	. . 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)))
53		prime 11786	. . . 4 ⊢ (𝑀 ∈ ℕ → (∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀)) ↔ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ 𝑗 ≤ 𝑀 ∧ (𝑀 / 𝑗) ∈ ℕ) → 𝑗 = 𝑀)))
54	53	adantl 475	. . 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)))
55	52, 54	sylibrd 251	. 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))))
56	18, 55	jcad 510	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 setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 1c1 10253 + caddc 10255 < clt 10391 ≤ cle 10392 / cdiv 11009 ℕcn 11350 ℕ₀cn0 11618 ℤcz 11704 !cfa 13353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-seq 13096 df-fac 13354

This theorem is referenced by: infpnlem2 15986

W3C validator