infpnlem1 - Intuitionistic Logic Explorer

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Theorem infpnlem1 12995

Description: Lemma for infpn 12997. 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 9542	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → 𝑁 ∈ ℤ)
3		nnz 9542	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
4	3	ad2antlr 489	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ 1 < 𝑀) → 𝑀 ∈ ℤ)
5		zdclt 9601	. . . . . 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 9192	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
8		nnre 9192	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
9		lenlt 8297	. . . . . . . 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 9451	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
13		facndiv 11047	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ) ∧ (1 < 𝑀 ∧ 𝑀 ≤ 𝑁)) → ¬ (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
14		infpnlem.1	. . . . . . . . . . 11 ⊢ 𝐾 = ((!‘𝑁) + 1)
15	14	oveq1i 6038	. . . . . . . . . 10 ⊢ (𝐾 / 𝑀) = (((!‘𝑁) + 1) / 𝑀)
16		nnz 9542	. . . . . . . . . 10 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (𝐾 / 𝑀) ∈ ℤ)
17	15, 16	eqeltrrid 2319	. . . . . . . . 9 ⊢ ((𝐾 / 𝑀) ∈ ℕ → (((!‘𝑁) + 1) / 𝑀) ∈ ℤ)
18	13, 17	nsyl 633	. . . . . . . 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 861	. . . . 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 11044	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ)
27	26	peano2nnd 9200	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ)
28	14, 27	eqeltrid 2318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ)
29	28	nncnd 9199	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℂ)
30		nndivtr 9227	. . . . . . . . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (((𝑀 / 𝑗) ∈ ℕ ∧ (𝐾 / 𝑀) ∈ ℕ) → (𝐾 / 𝑗) ∈ ℕ))
33	32	3expa 1230	. . . . . . . . . . . . . . . . . 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 9192	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
37		letri3 8302	. . . . . . . . . . . . . . . . . . . . . . . 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 1220	. . . . . . . 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 2600	. . . . . 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 9623	. . . 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 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 ∀wral 2511 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℂcc 8073 ℝcr 8074 1c1 8076 + caddc 8078 < clt 8256 ≤ cle 8257 / cdiv 8894 ℕcn 9185 ℕ₀cn0 9444 ℤcz 9523 !cfa 11033

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-n0 9445 df-z 9524 df-uz 9800 df-seqfrec 10756 df-fac 11034

This theorem is referenced by: infpnlem2 12996

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