rmxypairf1o - Mathbox for Stefan O'Rear

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Theorem rmxypairf1o 41635

Description: The function used to extract rational and irrational parts in df-rmx 41625 and df-rmy 41626 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 7438	. . . 4 ⊢ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ∈ V
2		eqid 2732	. . . 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 6690	. . 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 5952	. . 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 3478	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3478	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 7981	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1^st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7982	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2^nd ‘𝑏) = 𝑑)
10	9	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7423	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2743	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5840	. . . . . 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 2801	. . 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 2791	. 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 6888	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1^st ‘𝑏) = (1^st ‘𝑐))
19		fveq2 6888	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2^nd ‘𝑏) = (2^nd ‘𝑐))
20	19	oveq2d 7421	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐)))
21	18, 20	oveq12d 7423	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))))
22		ovex 7438	. . . . . . 7 ⊢ ((1^st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6995	. . . . . 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 726	. . . . 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 6888	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1^st ‘𝑏) = (1^st ‘𝑑))
26		fveq2 6888	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2^nd ‘𝑏) = (2^nd ‘𝑑))
27	26	oveq2d 7421	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑)))
28	25, 27	oveq12d 7423	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))))
29		ovex 7438	. . . . . . 7 ⊢ ((1^st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6995	. . . . . 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 727	. . . . 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 2748	. . . 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 41629	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
34	33	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35		nn0ssq 12937	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℚ
36		xp1st 8003	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑐) ∈ ℕ₀)
37	36	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℕ₀)
38	35, 37	sselid 3979	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℚ)
39		xp2nd 8004	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (2^nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑐) ∈ ℤ)
41		zq 12934	. . . . . . . 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 8003	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑑) ∈ ℕ₀)
44	43	ad2antll 727	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℕ₀)
45	35, 44	sselid 3979	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℚ)
46		xp2nd 8004	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ₀ × ℤ) → (2^nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 727	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑑) ∈ ℤ)
48		zq 12934	. . . . . . . 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 41631	. . . . . . 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 1379	. . . . . 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 8012	. . . . . 6 ⊢ ((𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ)) → (((1^st ‘𝑐) = (1^st ‘𝑑) ∧ (2^nd ‘𝑐) = (2^nd ‘𝑑)) ↔ 𝑐 = 𝑑))
54	53	adantl 482	. . . . 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 3200	. 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 7269	. 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 1343	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 396 = wceq 1541 ∈ wcel 2106 {cab 2709 ∀wral 3061 ∃wrex 3070 ∖ cdif 3944 ⟨cop 4633 ↦ cmpt 5230 × cxp 5673 ran crn 5676 Fn wfn 6535 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 ℂcc 11104 1c1 11107 + caddc 11109 · cmul 11111 − cmin 11440 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℚcq 12928 ↑cexp 14023 √csqrt 15176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-numer 16667 df-denom 16668

This theorem is referenced by: rmxyelxp 41636 rmxyval 41639

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