Theorem rmxypairf1o 43850

Description: The function used to extract rational and irrational parts in df-rmx 43841 and df-rmy 43842 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 7442	. . . 4 ⊢ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) ∈ V
2		eqid 2760	. . . 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 6671	. . 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 5936	. . 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 7995	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑏) = 𝑐)
9	6, 7	op2ndd 7996	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑏) = 𝑑)
10	9	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
11	8, 10	oveq12d 7427	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
12	11	eqeq2d 2771	. . . . . . 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 227	. . . . 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 2826	. . 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 2814	. 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 6874	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (1st ‘𝑏) = (1st ‘𝑐))
19		fveq2 6874	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (2nd ‘𝑏) = (2nd ‘𝑐))
20	19	oveq2d 7425	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐)))
21	18, 20	oveq12d 7427	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))))
22		ovex 7442	. . . . . . 7 ⊢ ((1st ‘𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑐))) ∈ V
23	21, 2, 22	fvmpt 6982	. . . . . 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 741	. . . . 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 6874	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (1st ‘𝑏) = (1st ‘𝑑))
26		fveq2 6874	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (2nd ‘𝑏) = (2nd ‘𝑑))
27	26	oveq2d 7425	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑)))
28	25, 27	oveq12d 7427	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))))
29		ovex 7442	. . . . . . 7 ⊢ ((1st ‘𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑑))) ∈ V
30	28, 2, 29	fvmpt 6982	. . . . . 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 742	. . . . 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 2776	. . . 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 43845	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
34	33	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35		nn0ssq 13039	. . . . . . . 8 ⊢ ℕ0 ⊆ ℚ
36		xp1st 8017	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (1st ‘𝑐) ∈ ℕ0)
37	36	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℕ0)
38	35, 37	sselid 3929	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑐) ∈ ℚ)
39		xp2nd 8018	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℕ0 × ℤ) → (2nd ‘𝑐) ∈ ℤ)
40	39	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑐) ∈ ℤ)
41		zq 13036	. . . . . . . 8 ⊢ ((2nd ‘𝑐) ∈ ℤ → (2nd ‘𝑐) ∈ ℚ)
42	40, 41	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑐) ∈ ℚ)
43		xp1st 8017	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (1st ‘𝑑) ∈ ℕ0)
44	43	ad2antll 742	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℕ0)
45	35, 44	sselid 3929	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st ‘𝑑) ∈ ℚ)
46		xp2nd 8018	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℕ0 × ℤ) → (2nd ‘𝑑) ∈ ℤ)
47	46	ad2antll 742	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑑) ∈ ℤ)
48		zq 13036	. . . . . . . 8 ⊢ ((2nd ‘𝑑) ∈ ℤ → (2nd ‘𝑑) ∈ ℚ)
49	47, 48	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd ‘𝑑) ∈ ℚ)
50		qirropth 43847	. . . . . . 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 1406	. . . . . 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 232	. . . . 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 8026	. . . . . 6 ⊢ ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑)) ↔ 𝑐 = 𝑑))
54	53	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st ‘𝑐) = (1st ‘𝑑) ∧ (2nd ‘𝑐) = (2nd ‘𝑑)) ↔ 𝑐 = 𝑑))
55	52, 54	sylibd 242	. . . 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 243	. . 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 3205	. 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 7272	. 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 1362	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))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ∖ cdif 3896  ⟨cop 4590   ↦ cmpt 5186   × cxp 5646  ran crn 5649   Fn wfn 6523  –1-1-onto→wf1o 6527  ‘cfv 6528  (class class class)co 7409  1^st c1st 7983  2^nd c2nd 7984  ℂcc 11155  1c1 11158   + caddc 11160   · cmul 11162   − cmin 11498  2c2 12352  ℕ₀cn0 12561  ℤcz 12648  ℤ_≥cuz 12920  ℚcq 13030  ↑cexp 14158  √csqrt 15353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-resscn 11214  ax-1cn 11215  ax-icn 11216  ax-addcl 11217  ax-addrcl 11218  ax-mulcl 11219  ax-mulrcl 11220  ax-mulcom 11221  ax-addass 11222  ax-mulass 11223  ax-distr 11224  ax-i2m1 11225  ax-1ne0 11226  ax-1rid 11227  ax-rnegex 11228  ax-rrecex 11229  ax-cnre 11230  ax-pre-lttri 11231  ax-pre-lttrn 11232  ax-pre-ltadd 11233  ax-pre-mulgt0 11234  ax-pre-sup 11235

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-inf 9413  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11500  df-neg 11501  df-div 11929  df-nn 12291  df-2 12360  df-3 12361  df-n0 12562  df-z 12649  df-uz 12921  df-q 13031  df-rp 13076  df-fl 13886  df-mod 13964  df-seq 14099  df-exp 14159  df-cj 15219  df-re 15220  df-im 15221  df-sqrt 15355  df-abs 15356  df-dvds 16376  df-gcd 16618  df-numer 16859  df-denom 16860

This theorem is used by:  rmxyelxp  43851  rmxyval  43854