4sqlem18 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 4sqlem18		GIF version

Theorem 4sqlem18 12806

Description: Lemma for 4sq 12808. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)

**Hypotheses**
Ref	Expression
4sqlem11.1	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
4sq.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
4sq.3	⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
4sq.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
4sq.5	⊢ (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)
4sq.6	⊢ 𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}
4sq.7	⊢ 𝑀 = inf(𝑇, ℝ, < )

**Assertion**
Ref	Expression
4sqlem18	⊢ (𝜑 → 𝑃 ∈ 𝑆)

Distinct variable groups: 𝑖,𝑀,𝑛 𝑛,𝑁 𝑃,𝑖,𝑛,𝑤,𝑥,𝑦,𝑧 𝑆,𝑖,𝑛 𝑇,𝑖 𝜑,𝑖,𝑛 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑤) 𝑆(𝑥,𝑦,𝑧,𝑤) 𝑇(𝑥,𝑦,𝑧,𝑤,𝑛) 𝑀(𝑥,𝑦,𝑧,𝑤) 𝑁(𝑥,𝑦,𝑧,𝑤,𝑖)

Proof of Theorem 4sqlem18 Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sq.4	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		prmnn 12507	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nncnd 9070	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℂ)
5	4	mullidd 8110	. 2 ⊢ (𝜑 → (1 · 𝑃) = 𝑃)
6		4sq.7	. . . . . . . . . . . 12 ⊢ 𝑀 = inf(𝑇, ℝ, < )
7		4sqlem11.1	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
8		4sq.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9		4sq.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
10		4sq.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)
11		4sq.6	. . . . . . . . . . . . . . 15 ⊢ 𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}
12	7, 8, 9, 1, 10, 11, 6	4sqlem13m 12801	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃))
13	12	simpld 112	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑗 𝑗 ∈ 𝑇)
14		1zzd 9419	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑇) → 1 ∈ ℤ)
15		nnuz 9704	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
16	15	rabeqi 2766	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆} = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
17	11, 16	eqtri 2227	. . . . . . . . . . . . . 14 ⊢ 𝑇 = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
18		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑇) → 𝑗 ∈ 𝑇)
19		elfznn 10196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑗) → 𝑖 ∈ ℕ)
20	19	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → 𝑖 ∈ ℕ)
21	3	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → 𝑃 ∈ ℕ)
22	20, 21	nnmulcld 9105	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → (𝑖 · 𝑃) ∈ ℕ)
23	22	nnnn0d 9368	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → (𝑖 · 𝑃) ∈ ℕ₀)
24	7	4sqlemsdc 12798	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 · 𝑃) ∈ ℕ₀ → DECID (𝑖 · 𝑃) ∈ 𝑆)
25	23, 24	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → DECID (𝑖 · 𝑃) ∈ 𝑆)
26	14, 17, 18, 25	infssuzcldc 10400	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑇) → inf(𝑇, ℝ, < ) ∈ 𝑇)
27	13, 26	exlimddv 1923	. . . . . . . . . . . 12 ⊢ (𝜑 → inf(𝑇, ℝ, < ) ∈ 𝑇)
28	6, 27	eqeltrid 2293	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ 𝑇)
29		oveq1 5964	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑀 → (𝑖 · 𝑃) = (𝑀 · 𝑃))
30	29	eleq1d 2275	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑀 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (𝑀 · 𝑃) ∈ 𝑆))
31	30, 11	elrab2 2936	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝑇 ↔ (𝑀 ∈ ℕ ∧ (𝑀 · 𝑃) ∈ 𝑆))
32	28, 31	sylib 122	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ (𝑀 · 𝑃) ∈ 𝑆))
33	32	simprd 114	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ 𝑆)
34	7	4sqlem2 12787	. . . . . . . . 9 ⊢ ((𝑀 · 𝑃) ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
35	33, 34	sylib 122	. . . . . . . 8 ⊢ (𝜑 → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
36	35	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
37		simp1l 1024	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝜑)
38	37, 8	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑁 ∈ ℕ)
39	37, 9	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑃 = ((2 · 𝑁) + 1))
40	37, 1	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑃 ∈ ℙ)
41	37, 10	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → (0...(2 · 𝑁)) ⊆ 𝑆)
42		simp1r 1025	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑀 ∈ (ℤ_≥‘2))
43		simp2ll 1067	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑎 ∈ ℤ)
44		simp2lr 1068	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑏 ∈ ℤ)
45		simp2rl 1069	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑐 ∈ ℤ)
46		simp2rr 1070	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → 𝑑 ∈ ℤ)
47		eqid 2206	. . . . . . . . . . . . 13 ⊢ (((𝑎 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑎 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
48		eqid 2206	. . . . . . . . . . . . 13 ⊢ (((𝑏 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑏 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
49		eqid 2206	. . . . . . . . . . . . 13 ⊢ (((𝑐 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑐 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
50		eqid 2206	. . . . . . . . . . . . 13 ⊢ (((𝑑 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑑 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
51		eqid 2206	. . . . . . . . . . . . 13 ⊢ (((((((𝑎 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2) + ((((𝑏 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2)) + (((((𝑐 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2) + ((((𝑑 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2))) / 𝑀) = (((((((𝑎 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2) + ((((𝑏 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2)) + (((((𝑐 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2) + ((((𝑑 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))↑2))) / 𝑀)
52		simp3 1002	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
53	7, 38, 39, 40, 41, 11, 6, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52	4sqlem17 12805	. . . . . . . . . . . 12 ⊢ ¬ ((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
54	53	pm2.21i 647	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) ∧ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))) → ¬ 𝑀 ∈ (ℤ_≥‘2))
55	54	3expia 1208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ))) → ((𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → ¬ 𝑀 ∈ (ℤ_≥‘2)))
56	55	anassrs 400	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → ((𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → ¬ 𝑀 ∈ (ℤ_≥‘2)))
57	56	rexlimdvva 2632	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → ¬ 𝑀 ∈ (ℤ_≥‘2)))
58	57	rexlimdvva 2632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ (𝑀 · 𝑃) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → ¬ 𝑀 ∈ (ℤ_≥‘2)))
59	36, 58	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ_≥‘2)) → ¬ 𝑀 ∈ (ℤ_≥‘2))
60	59	pm2.01da 637	. . . . 5 ⊢ (𝜑 → ¬ 𝑀 ∈ (ℤ_≥‘2))
61	32	simpld 112	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
62		elnn1uz2 9748	. . . . . 6 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ_≥‘2)))
63	61, 62	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑀 = 1 ∨ 𝑀 ∈ (ℤ_≥‘2)))
64	60, 63	ecased 1362	. . . 4 ⊢ (𝜑 → 𝑀 = 1)
65	64, 28	eqeltrrd 2284	. . 3 ⊢ (𝜑 → 1 ∈ 𝑇)
66		oveq1 5964	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 · 𝑃) = (1 · 𝑃))
67	66	eleq1d 2275	. . . . 5 ⊢ (𝑖 = 1 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (1 · 𝑃) ∈ 𝑆))
68	67, 11	elrab2 2936	. . . 4 ⊢ (1 ∈ 𝑇 ↔ (1 ∈ ℕ ∧ (1 · 𝑃) ∈ 𝑆))
69	68	simprbi 275	. . 3 ⊢ (1 ∈ 𝑇 → (1 · 𝑃) ∈ 𝑆)
70	65, 69	syl 14	. 2 ⊢ (𝜑 → (1 · 𝑃) ∈ 𝑆)
71	5, 70	eqeltrrd 2284	1 ⊢ (𝜑 → 𝑃 ∈ 𝑆)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 710 DECID wdc 836 ∧ w3a 981 = wceq 1373 ∃wex 1516 ∈ wcel 2177 {cab 2192 ∃wrex 2486 {crab 2489 ⊆ wss 3170 class class class wbr 4051 ‘cfv 5280 (class class class)co 5957 infcinf 7100 ℝcr 7944 0cc0 7945 1c1 7946 + caddc 7948 · cmul 7950 < clt 8127 − cmin 8263 / cdiv 8765 ℕcn 9056 2c2 9107 ℕ₀cn0 9315 ℤcz 9392 ℤ_≥cuz 9668 ...cfz 10150 mod cmo 10489 ↑cexp 10705 ℙcprime 12504

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 ax-arch 8064 ax-caucvg 8065

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-id 4348 df-po 4351 df-iso 4352 df-iord 4421 df-on 4423 df-ilim 4424 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-isom 5289 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-recs 6404 df-irdg 6469 df-frec 6490 df-1o 6515 df-2o 6516 df-oadd 6519 df-er 6633 df-en 6841 df-dom 6842 df-fin 6843 df-sup 7101 df-inf 7102 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 df-inn 9057 df-2 9115 df-3 9116 df-4 9117 df-n0 9316 df-z 9393 df-uz 9669 df-q 9761 df-rp 9796 df-fz 10151 df-fzo 10285 df-fl 10435 df-mod 10490 df-seqfrec 10615 df-exp 10706 df-ihash 10943 df-cj 11228 df-re 11229 df-im 11230 df-rsqrt 11384 df-abs 11385 df-dvds 12174 df-gcd 12350 df-prm 12505 df-gz 12768

This theorem is referenced by: 4sqlem19 12807

W3C validator