rmxypairf1o - Mathbox for Stefan O'Rear

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Theorem rmxypairf1o 41953

Description: The function used to extract rational and irrational parts in df-rmx 41943 and df-rmy 41944 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 7445	. . . 4 ⊢ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ∈ V
2		eqid 2731	. . . 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 6694	. . 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 5955	. . 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 3477	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3477	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 7988	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1^st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7989	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2^nd ‘𝑏) = 𝑑)
10	9	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7430	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2742	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5843	. . . . . 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 223	. . . . 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 2800	. . 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 6892	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1^st ‘𝑏) = (1^st ‘𝑐))
19		fveq2 6892	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2^nd ‘𝑏) = (2^nd ‘𝑐))
20	19	oveq2d 7428	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐)))
21	18, 20	oveq12d 7430	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))))
22		ovex 7445	. . . . . . 7 ⊢ ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6999	. . . . . 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 725	. . . . 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 6892	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1^st ‘𝑏) = (1^st ‘𝑑))
26		fveq2 6892	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2^nd ‘𝑏) = (2^nd ‘𝑑))
27	26	oveq2d 7428	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑)))
28	25, 27	oveq12d 7430	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))))
29		ovex 7445	. . . . . . 7 ⊢ ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6999	. . . . . 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 726	. . . . 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 2747	. . . 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 41947	. . . . . . . 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 12946	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℚ
36		xp1st 8010	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑐) ∈ ℕ₀)
37	36	ad2antrl 725	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℕ₀)
38	35, 37	sselid 3981	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℚ)
39		xp2nd 8011	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (2^nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 725	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑐) ∈ ℤ)
41		zq 12943	. . . . . . . 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 8010	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑑) ∈ ℕ₀)
44	43	ad2antll 726	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℕ₀)
45	35, 44	sselid 3981	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℚ)
46		xp2nd 8011	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → (2^nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 726	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑑) ∈ ℤ)
48		zq 12943	. . . . . . . 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 41949	. . . . . . 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 1378	. . . . . 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 228	. . . . 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 8019	. . . . . 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 238	. . . 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 239	. . 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 3199	. 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 7276	. 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 1342	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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 ∃wrex 3069 ∖ cdif 3946 ⟨cop 4635 ↦ cmpt 5232 × cxp 5675 ran crn 5678 Fn wfn 6539 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7412 1^st c1st 7976 2^nd c2nd 7977 ℂcc 11111 1c1 11114 + caddc 11116 · cmul 11118 − cmin 11449 2c2 12272 ℕ₀cn0 12477 ℤcz 12563 ℤ_≥cuz 12827 ℚcq 12937 ↑cexp 14032 √csqrt 15185

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-dvds 16203 df-gcd 16441 df-numer 16676 df-denom 16677

This theorem is referenced by: rmxyelxp 41954 rmxyval 41957

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