Theorem rmxypairf1o 43297

Description: The function used to extract rational and irrational parts in df-rmx 43288 and df-rmy 43289 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) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘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 7403	. . . 4 ⊢ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ∈ V
2		eqid 2737	. . . 4 ⊢ (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))
3	1, 2	fnmpti 6645	. . 3 ⊢ (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) Fn (ℕ0 × ℤ)
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) Fn (ℕ0 × ℤ))
5	2	rnmpt 5916	. . 3 ⊢ ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))}
6		vex 3446	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3446	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 7955	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7956	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑏) = 𝑑)
10	9	oveq2d 7386	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7388	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2748	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5801	. . . . . 6 ⊢ (∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘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 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))
15	14	a1i 11	. . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))))
16	15	abbidv 2803	. . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))})
17	5, 16	eqtr4id 2791	. 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) = {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
18		fveq2 6844	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1st ‘𝑏) = (1st ‘𝑐))
19		fveq2 6844	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2nd ‘𝑏) = (2nd ‘𝑐))
20	19	oveq2d 7386	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐)))
21	18, 20	oveq12d 7388	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))))
22		ovex 7403	. . . . . . 7 ⊢ ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6951	. . . . . 6 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑐) = ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))))
24	23	ad2antrl 729	. . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑐) = ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))))
25		fveq2 6844	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1st ‘𝑏) = (1st ‘𝑑))
26		fveq2 6844	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2nd ‘𝑏) = (2nd ‘𝑑))
27	26	oveq2d 7386	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑)))
28	25, 27	oveq12d 7388	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))))
29		ovex 7403	. . . . . . 7 ⊢ ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6951	. . . . . 6 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑑) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))))
31	30	ad2antll 730	. . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑑) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))))
32	24, 31	eqeq12d 2753	. . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑑) ↔ ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑)))))
33		rmspecsqrtnq 43292	. . . . . . . 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 12884	. . . . . . . 8 ⊢ ℕ0 ⊆ ℚ
36		xp1st 7977	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (1st ‘𝑐) ∈ ℕ0)
37	36	ad2antrl 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℕ0)
38	35, 37	sselid 3933	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℚ)
39		xp2nd 7978	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (2nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑐) ∈ ℤ)
41		zq 12881	. . . . . . . 8 ⊢ ((2nd ‘𝑐) ∈ ℤ → (2nd ‘𝑐) ∈ ℚ)
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑐) ∈ ℚ)
43		xp1st 7977	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (1st ‘𝑑) ∈ ℕ0)
44	43	ad2antll 730	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℕ0)
45	35, 44	sselid 3933	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℚ)
46		xp2nd 7978	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (2nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 730	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑑) ∈ ℤ)
48		zq 12881	. . . . . . . 8 ⊢ ((2nd ‘𝑑) ∈ ℤ → (2nd ‘𝑑) ∈ ℚ)
49	47, 48	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑑) ∈ ℚ)
50		qirropth 43294	. . . . . . 7 ⊢ (((√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ) ∧ ((1st ‘𝑐) ∈ ℚ ∧ (2nd ‘𝑐) ∈ ℚ) ∧ ((1st ‘𝑑) ∈ ℚ ∧ (2nd ‘𝑑) ∈ ℚ)) → (((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) ↔ ((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑))))
51	34, 38, 42, 45, 49, 50	syl122anc 1382	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) ↔ ((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑))))
52	51	biimpd 229	. . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) → ((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑))))
53		xpopth 7986	. . . . . 6 ⊢ ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑)) ↔ 𝑐 = 𝑑))
54	53	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑)) ↔ 𝑐 = 𝑑))
55	52, 54	sylibd 239	. . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) → 𝑐 = 𝑑))
56	32, 55	sylbid 240	. . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑑) → 𝑐 = 𝑑))
57	56	ralrimivva 3181	. 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ∀𝑐 ∈ (ℕ0 × ℤ)∀𝑑 ∈ (ℕ0 × ℤ)(((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑑) → 𝑐 = 𝑑))
58		dff1o6 7233	. 2 ⊢ ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ↔ ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) Fn (ℕ0 × ℤ) ∧ ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))) = {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ∧ ∀𝑐 ∈ (ℕ0 × ℤ)∀𝑑 ∈ (ℕ0 × ℤ)(((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘𝑑) → 𝑐 = 𝑑)))
59	4, 17, 57, 58	syl3anbrc 1345	1 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∖ cdif 3900  ⟨cop 4588   ↦ cmpt 5181   × cxp 5632  ran crn 5635   Fn wfn 6497  –1-1-onto→wf1o 6501  ‘cfv 6502  (class class class)co 7370  1^st c1st 7943  2^nd c2nd 7944  ℂcc 11038  1c1 11041   + caddc 11043   · cmul 11045   − cmin 11378  2c2 12214  ℕ₀cn0 12415  ℤcz 12502  ℤ_≥cuz 12765  ℚcq 12875  ↑cexp 13998  √csqrt 15170

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 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-sup 9359  df-inf 9360  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-n0 12416  df-z 12503  df-uz 12766  df-q 12876  df-rp 12920  df-fl 13726  df-mod 13804  df-seq 13939  df-exp 13999  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-dvds 16194  df-gcd 16436  df-numer 16676  df-denom 16677

This theorem is referenced by:  rmxyelxp  43298  rmxyval  43301