Theorem rmxypairf1o 43493

Description: The function used to extract rational and irrational parts in df-rmx 43484 and df-rmy 43485 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 7431	. . . 4 ⊢ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ∈ V
2		eqid 2764	. . . 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 6666	. . 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 5935	. . 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 3460	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
7		vex 3460	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
8	6, 7	op1std 7982	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7983	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑏) = 𝑑)
10	9	oveq2d 7414	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7416	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2775	. . . . . . 7 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
13	12	rexxp 5816	. . . . . 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 226	. . . . 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 2830	. . 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 2818	. 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 6869	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1st ‘𝑏) = (1st ‘𝑐))
19		fveq2 6869	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2nd ‘𝑏) = (2nd ‘𝑐))
20	19	oveq2d 7414	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐)))
21	18, 20	oveq12d 7416	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))))
22		ovex 7431	. . . . . . 7 ⊢ ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6977	. . . . . 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 738	. . . . 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 6869	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1st ‘𝑏) = (1st ‘𝑑))
26		fveq2 6869	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2nd ‘𝑏) = (2nd ‘𝑑))
27	26	oveq2d 7414	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑)))
28	25, 27	oveq12d 7416	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))))
29		ovex 7431	. . . . . . 7 ⊢ ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6977	. . . . . 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 739	. . . . 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 2780	. . . 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 43488	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
34	33	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35		nn0ssq 12960	. . . . . . . 8 ⊢ ℕ0 ⊆ ℚ
36		xp1st 8004	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (1st ‘𝑐) ∈ ℕ0)
37	36	ad2antrl 738	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℕ0)
38	35, 37	sselid 3936	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℚ)
39		xp2nd 8005	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (2nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 738	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑐) ∈ ℤ)
41		zq 12957	. . . . . . . 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 8004	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (1st ‘𝑑) ∈ ℕ0)
44	43	ad2antll 739	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℕ0)
45	35, 44	sselid 3936	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℚ)
46		xp2nd 8005	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (2nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 739	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑑) ∈ ℤ)
48		zq 12957	. . . . . . . 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 43490	. . . . . . 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 1400	. . . . . 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 231	. . . . 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 8013	. . . . . 6 ⊢ ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑)) ↔ 𝑐 = 𝑑))
54	53	adantl 485	. . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑)) ↔ 𝑐 = 𝑑))
55	52, 54	sylibd 241	. . . 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 242	. . 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 3207	. 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 7261	. 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 1358	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 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903  ⟨cop 4590   ↦ cmpt 5183   × cxp 5647  ran crn 5650   Fn wfn 6518  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  ℂcc 11073  1c1 11076   + caddc 11078   · cmul 11080   − cmin 11416  2c2 12274  ℕ₀cn0 12483  ℤcz 12570  ℤ_≥cuz 12841  ℚcq 12951  ↑cexp 14076  √csqrt 15262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  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 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-sup 9390  df-inf 9391  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-q 12952  df-rp 12996  df-fl 13804  df-mod 13882  df-seq 14017  df-exp 14077  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-dvds 16289  df-gcd 16531  df-numer 16772  df-denom 16773

This theorem is referenced by:  rmxyelxp  43494  rmxyval  43497