rmxypairf1o - Mathbox for Stefan O'Rear

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Theorem rmxypairf1o 43153

Description: The function used to extract rational and irrational parts in df-rmx 43144 and df-rmy 43145 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)

**Assertion**
Ref	Expression
rmxypairf1o	⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))):(ℕ₀ × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Distinct variable group: 𝑏,𝑐,𝑑,𝑎,𝐴

**Proof of Theorem rmxypairf1o**
Step	Hyp	Ref	Expression
1		ovex 7391	. . . 4 ⊢ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ∈ V
2		eqid 2736	. . . 4 ⊢ (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) = (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))
3	1, 2	fnmpti 6635	. . 3 ⊢ (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) Fn (ℕ₀ × ℤ)
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) Fn (ℕ₀ × ℤ))
5	2	rnmpt 5906	. . 3 ⊢ ran (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ₀ × ℤ)𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))}
6		vex 3444	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3444	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 7943	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1^st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7944	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2^nd ‘𝑏) = 𝑑)
10	9	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7376	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2747	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5791	. . . . . 6 ⊢ (∃𝑏 ∈ (ℕ₀ × ℤ)𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ↔ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
14	13	bicomi 224	. . . . 5 ⊢ (∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ₀ × ℤ)𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))
15	14	a1i 11	. . . 4 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ₀ × ℤ)𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
16	15	abbidv 2802	. . 3 ⊢ (𝐴 ∈ (ℤ_≥‘2) → {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} = {𝑎 ∣ ∃𝑏 ∈ (ℕ₀ × ℤ)𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))})
17	5, 16	eqtr4id 2790	. 2 ⊢ (𝐴 ∈ (ℤ_≥‘2) → ran (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) = {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
18		fveq2 6834	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1^st ‘𝑏) = (1^st ‘𝑐))
19		fveq2 6834	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2^nd ‘𝑏) = (2^nd ‘𝑐))
20	19	oveq2d 7374	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐)))
21	18, 20	oveq12d 7376	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))))
22		ovex 7391	. . . . . . 7 ⊢ ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6941	. . . . . 6 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑐) = ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))))
24	23	ad2antrl 728	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑐) = ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))))
25		fveq2 6834	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1^st ‘𝑏) = (1^st ‘𝑑))
26		fveq2 6834	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2^nd ‘𝑏) = (2^nd ‘𝑑))
27	26	oveq2d 7374	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑)))
28	25, 27	oveq12d 7376	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))))
29		ovex 7391	. . . . . . 7 ⊢ ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6941	. . . . . 6 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑑) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))))
31	30	ad2antll 729	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑑) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))))
32	24, 31	eqeq12d 2752	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑑) ↔ ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑)))))
33		rmspecsqrtnq 43148	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
34	33	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35		nn0ssq 12870	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℚ
36		xp1st 7965	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑐) ∈ ℕ₀)
37	36	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℕ₀)
38	35, 37	sselid 3931	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℚ)
39		xp2nd 7966	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (2^nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑐) ∈ ℤ)
41		zq 12867	. . . . . . . 8 ⊢ ((2^nd ‘𝑐) ∈ ℤ → (2^nd ‘𝑐) ∈ ℚ)
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑐) ∈ ℚ)
43		xp1st 7965	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑑) ∈ ℕ₀)
44	43	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℕ₀)
45	35, 44	sselid 3931	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℚ)
46		xp2nd 7966	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → (2^nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑑) ∈ ℤ)
48		zq 12867	. . . . . . . 8 ⊢ ((2^nd ‘𝑑) ∈ ℤ → (2^nd ‘𝑑) ∈ ℚ)
49	47, 48	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑑) ∈ ℚ)
50		qirropth 43150	. . . . . . 7 ⊢ (((√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ) ∧ ((1^st ‘𝑐) ∈ ℚ ∧ (2^nd ‘𝑐) ∈ ℚ) ∧ ((1^st ‘𝑑) ∈ ℚ ∧ (2^nd ‘𝑑) ∈ ℚ)) → (((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) ↔ ((1^st ‘𝑐) = (1^st ‘𝑑) ∧ (2^nd ‘𝑐) = (2^nd ‘𝑑))))
51	34, 38, 42, 45, 49, 50	syl122anc 1381	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) ↔ ((1^st ‘𝑐) = (1^st ‘𝑑) ∧ (2^nd ‘𝑐) = (2^nd ‘𝑑))))
52	51	biimpd 229	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) → ((1^st ‘𝑐) = (1^st ‘𝑑) ∧ (2^nd ‘𝑐) = (2^nd ‘𝑑))))
53		xpopth 7974	. . . . . 6 ⊢ ((𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ)) → (((1^st ‘𝑐) = (1^st ‘𝑑) ∧ (2^nd ‘𝑐) = (2^nd ‘𝑑)) ↔ 𝑐 = 𝑑))
54	53	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (((1^st ‘𝑐) = (1^st ‘𝑑) ∧ (2^nd ‘𝑐) = (2^nd ‘𝑑)) ↔ 𝑐 = 𝑑))
55	52, 54	sylibd 239	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) → 𝑐 = 𝑑))
56	32, 55	sylbid 240	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑑) → 𝑐 = 𝑑))
57	56	ralrimivva 3179	. 2 ⊢ (𝐴 ∈ (ℤ_≥‘2) → ∀𝑐 ∈ (ℕ₀ × ℤ)∀𝑑 ∈ (ℕ₀ × ℤ)(((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑑) → 𝑐 = 𝑑))
58		dff1o6 7221	. 2 ⊢ ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))):(ℕ₀ × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ↔ ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) Fn (ℕ₀ × ℤ) ∧ ran (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) = {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ∧ ∀𝑐 ∈ (ℕ₀ × ℤ)∀𝑑 ∈ (ℕ₀ × ℤ)(((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘𝑑) → 𝑐 = 𝑑)))
59	4, 17, 57, 58	syl3anbrc 1344	1 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))):(ℕ₀ × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {cab 2714 ∀wral 3051 ∃wrex 3060 ∖ cdif 3898 ⟨cop 4586 ↦ cmpt 5179 × cxp 5622 ran crn 5625 Fn wfn 6487 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 ℂcc 11024 1c1 11027 + caddc 11029 · cmul 11031 − cmin 11364 2c2 12200 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 ℚcq 12861 ↑cexp 13984 √csqrt 15156

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-gcd 16422 df-numer 16662 df-denom 16663

This theorem is referenced by: rmxyelxp 43154 rmxyval 43157

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