Theorem rmxypairf1o 42907

Description: The function used to extract rational and irrational parts in df-rmx 42897 and df-rmy 42898 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 7423	. . . 4 ⊢ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ∈ V
2		eqid 2730	. . . 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 6664	. . 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 5924	. . 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 3454	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3454	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 7981	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7982	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑏) = 𝑑)
10	9	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7408	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2741	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5809	. . . . . 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 2796	. . 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 2784	. 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 6861	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1st ‘𝑏) = (1st ‘𝑐))
19		fveq2 6861	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2nd ‘𝑏) = (2nd ‘𝑐))
20	19	oveq2d 7406	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐)))
21	18, 20	oveq12d 7408	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))))
22		ovex 7423	. . . . . . 7 ⊢ ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6971	. . . . . 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 728	. . . . 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 6861	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1st ‘𝑏) = (1st ‘𝑑))
26		fveq2 6861	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2nd ‘𝑏) = (2nd ‘𝑑))
27	26	oveq2d 7406	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑)))
28	25, 27	oveq12d 7408	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))))
29		ovex 7423	. . . . . . 7 ⊢ ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6971	. . . . . 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 729	. . . . 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 2746	. . . 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 42901	. . . . . . . 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 12923	. . . . . . . 8 ⊢ ℕ0 ⊆ ℚ
36		xp1st 8003	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (1st ‘𝑐) ∈ ℕ0)
37	36	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℕ0)
38	35, 37	sselid 3947	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℚ)
39		xp2nd 8004	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (2nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑐) ∈ ℤ)
41		zq 12920	. . . . . . . 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 8003	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (1st ‘𝑑) ∈ ℕ0)
44	43	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℕ0)
45	35, 44	sselid 3947	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℚ)
46		xp2nd 8004	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (2nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑑) ∈ ℤ)
48		zq 12920	. . . . . . . 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 42903	. . . . . . 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 1381	. . . . . 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 8012	. . . . . 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 7253	. 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 1344	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 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914  ⟨cop 4598   ↦ cmpt 5191   × cxp 5639  ran crn 5642   Fn wfn 6509  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  ℂcc 11073  1c1 11076   + caddc 11078   · cmul 11080   − cmin 11412  2c2 12248  ℕ₀cn0 12449  ℤcz 12536  ℤ_≥cuz 12800  ℚcq 12914  ↑cexp 14033  √csqrt 15206

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-fl 13761  df-mod 13839  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16230  df-gcd 16472  df-numer 16712  df-denom 16713

This theorem is referenced by:  rmxyelxp  42908  rmxyval  42911