rmxypairf1o - Mathbox for Stefan O'Rear

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Theorem rmxypairf1o 42923

Description: The function used to extract rational and irrational parts in df-rmx 42913 and df-rmy 42914 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 7464	. . . 4 ⊢ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ∈ V
2		eqid 2737	. . . 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 6711	. . 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 5968	. . 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 3484	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3484	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 8024	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1^st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 8025	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2^nd ‘𝑏) = 𝑑)
10	9	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7449	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2748	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5853	. . . . . 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 2808	. . 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 2796	. 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 6906	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1^st ‘𝑏) = (1^st ‘𝑐))
19		fveq2 6906	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2^nd ‘𝑏) = (2^nd ‘𝑐))
20	19	oveq2d 7447	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐)))
21	18, 20	oveq12d 7449	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))))
22		ovex 7464	. . . . . . 7 ⊢ ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 7016	. . . . . 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 6906	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1^st ‘𝑏) = (1^st ‘𝑑))
26		fveq2 6906	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2^nd ‘𝑏) = (2^nd ‘𝑑))
27	26	oveq2d 7447	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑)))
28	25, 27	oveq12d 7449	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))))
29		ovex 7464	. . . . . . 7 ⊢ ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 7016	. . . . . 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 2753	. . . 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 42917	. . . . . . . 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 12999	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℚ
36		xp1st 8046	. . . . . . . . 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 3981	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℚ)
39		xp2nd 8047	. . . . . . . . 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 12996	. . . . . . . 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 8046	. . . . . . . . 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 3981	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℚ)
46		xp2nd 8047	. . . . . . . . 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 12996	. . . . . . . 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 42919	. . . . . . 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 8055	. . . . . 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 3202	. 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 7295	. 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 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ∖ cdif 3948 ⟨cop 4632 ↦ cmpt 5225 × cxp 5683 ran crn 5686 Fn wfn 6556 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ℚcq 12990 ↑cexp 14102 √csqrt 15272

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-numer 16772 df-denom 16773

This theorem is referenced by: rmxyelxp 42924 rmxyval 42927

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