rmxypairf1o - Mathbox for Stefan O'Rear

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Theorem rmxypairf1o 43190

Description: The function used to extract rational and irrational parts in df-rmx 43181 and df-rmy 43182 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 2735	. . . 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 6634	. . 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 5905	. . 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 3443	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3443	. . . . . . . . . 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 2746	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5790	. . . . . 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 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 2789	. 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 6833	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1^st ‘𝑏) = (1^st ‘𝑐))
19		fveq2 6833	. . . . . . . . 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 6940	. . . . . 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 729	. . . . 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 6833	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1^st ‘𝑏) = (1^st ‘𝑑))
26		fveq2 6833	. . . . . . . . 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 6940	. . . . . 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 730	. . . . 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 2751	. . . 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 43185	. . . . . . . 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 12872	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℚ
36		xp1st 7965	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ₀ × ℤ) → (1^st ‘𝑐) ∈ ℕ₀)
37	36	ad2antrl 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑐) ∈ ℕ₀)
38	35, 37	sselid 3930	. . . . . . 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 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑐) ∈ ℤ)
41		zq 12869	. . . . . . . 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 730	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (1^st ‘𝑑) ∈ ℕ₀)
45	35, 44	sselid 3930	. . . . . . 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 730	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑐 ∈ (ℕ₀ × ℤ) ∧ 𝑑 ∈ (ℕ₀ × ℤ))) → (2^nd ‘𝑑) ∈ ℤ)
48		zq 12869	. . . . . . . 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 43187	. . . . . . 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 1382	. . . . . 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 3178	. 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 1345	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 1542 ∈ wcel 2114 {cab 2713 ∀wral 3050 ∃wrex 3059 ∖ cdif 3897 ⟨cop 4585 ↦ cmpt 5178 × cxp 5621 ran crn 5624 Fn wfn 6486 –1-1-onto→wf1o 6490 ‘cfv 6491 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 ℂcc 11026 1c1 11029 + caddc 11031 · cmul 11033 − cmin 11366 2c2 12202 ℕ₀cn0 12403 ℤcz 12490 ℤ_≥cuz 12753 ℚcq 12863 ↑cexp 13986 √csqrt 15158

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 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 8886 df-dom 8887 df-sdom 8888 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-q 12864 df-rp 12908 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424 df-numer 16664 df-denom 16665

This theorem is referenced by: rmxyelxp 43191 rmxyval 43194

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