4sqlem18 - Intuitionistic Logic Explorer

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Theorem 4sqlem18 12546

Description: Lemma for 4sq 12548. 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 12248	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nncnd 8996	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℂ)
5	4	mullidd 8037	. 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 12541	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃))
13	12	simpld 112	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑗 𝑗 ∈ 𝑇)
14		1zzd 9344	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑇) → 1 ∈ ℤ)
15		nnuz 9628	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
16	15	rabeqi 2753	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆} = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
17	11, 16	eqtri 2214	. . . . . . . . . . . . . 14 ⊢ 𝑇 = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
18		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑇) → 𝑗 ∈ 𝑇)
19		elfznn 10120	. . . . . . . . . . . . . . . . . 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 9031	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → (𝑖 · 𝑃) ∈ ℕ)
23	22	nnnn0d 9293	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑗)) → (𝑖 · 𝑃) ∈ ℕ₀)
24	7	4sqlemsdc 12538	. . . . . . . . . . . . . . 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 12088	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑇) → inf(𝑇, ℝ, < ) ∈ 𝑇)
27	13, 26	exlimddv 1910	. . . . . . . . . . . 12 ⊢ (𝜑 → inf(𝑇, ℝ, < ) ∈ 𝑇)
28	6, 27	eqeltrid 2280	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ 𝑇)
29		oveq1 5925	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑀 → (𝑖 · 𝑃) = (𝑀 · 𝑃))
30	29	eleq1d 2262	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑀 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (𝑀 · 𝑃) ∈ 𝑆))
31	30, 11	elrab2 2919	. . . . . . . . . . 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 12527	. . . . . . . . 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 1023	. . . . . . . . . . . . . 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 1024	. . . . . . . . . . . . 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 1066	. . . . . . . . . . . . 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 1067	. . . . . . . . . . . . 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 1068	. . . . . . . . . . . . 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 1069	. . . . . . . . . . . . 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 2193	. . . . . . . . . . . . 13 ⊢ (((𝑎 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑎 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
48		eqid 2193	. . . . . . . . . . . . 13 ⊢ (((𝑏 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑏 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
49		eqid 2193	. . . . . . . . . . . . 13 ⊢ (((𝑐 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑐 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
50		eqid 2193	. . . . . . . . . . . . 13 ⊢ (((𝑑 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑑 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
51		eqid 2193	. . . . . . . . . . . . 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 1001	. . . . . . . . . . . . 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 12545	. . . . . . . . . . . 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 1207	. . . . . . . . . 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 2619	. . . . . . . 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 2619	. . . . . . 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 9672	. . . . . 6 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ_≥‘2)))
63	61, 62	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑀 = 1 ∨ 𝑀 ∈ (ℤ_≥‘2)))
64	60, 63	ecased 1360	. . . 4 ⊢ (𝜑 → 𝑀 = 1)
65	64, 28	eqeltrrd 2271	. . 3 ⊢ (𝜑 → 1 ∈ 𝑇)
66		oveq1 5925	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 · 𝑃) = (1 · 𝑃))
67	66	eleq1d 2262	. . . . 5 ⊢ (𝑖 = 1 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (1 · 𝑃) ∈ 𝑆))
68	67, 11	elrab2 2919	. . . 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 2271	1 ⊢ (𝜑 → 𝑃 ∈ 𝑆)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2164 {cab 2179 ∃wrex 2473 {crab 2476 ⊆ wss 3153 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 infcinf 7042 ℝcr 7871 0cc0 7872 1c1 7873 + caddc 7875 · cmul 7877 < clt 8054 − cmin 8190 / cdiv 8691 ℕcn 8982 2c2 9033 ℕ₀cn0 9240 ℤcz 9317 ℤ_≥cuz 9592 ...cfz 10074 mod cmo 10393 ↑cexp 10609 ℙcprime 12245

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 ax-arch 7991 ax-caucvg 7992

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-if 3558 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-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-2o 6470 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 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-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080 df-prm 12246 df-gz 12508

This theorem is referenced by: 4sqlem19 12547

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