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Theorem rmxypairf1o 36995
Description: The function used to extract rational and irrational parts in df-rmx 36985 and df-rmy 36986 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) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
Distinct variable group:   𝑏,𝑐,𝑑,𝑎,𝐴

Proof of Theorem rmxypairf1o
StepHypRef Expression
1 ovex 6643 . . . 4 ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ∈ V
2 eqid 2621 . . . 4 (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
31, 2fnmpti 5989 . . 3 (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) Fn (ℕ0 × ℤ)
43a1i 11 . 2 (𝐴 ∈ (ℤ‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) Fn (ℕ0 × ℤ))
5 vex 3193 . . . . . . . . . 10 𝑐 ∈ V
6 vex 3193 . . . . . . . . . 10 𝑑 ∈ V
75, 6op1std 7138 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → (1st𝑏) = 𝑐)
85, 6op2ndd 7139 . . . . . . . . . 10 (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd𝑏) = 𝑑)
98oveq2d 6631 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
107, 9oveq12d 6633 . . . . . . . 8 (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1110eqeq2d 2631 . . . . . . 7 (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
1211rexxp 5234 . . . . . 6 (∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1312bicomi 214 . . . . 5 (∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
1413a1i 11 . . . 4 (𝐴 ∈ (ℤ‘2) → (∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
1514abbidv 2738 . . 3 (𝐴 ∈ (ℤ‘2) → {𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))})
162rnmpt 5341 . . 3 ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))}
1715, 16syl6reqr 2674 . 2 (𝐴 ∈ (ℤ‘2) → ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
18 fveq2 6158 . . . . . . . 8 (𝑏 = 𝑐 → (1st𝑏) = (1st𝑐))
19 fveq2 6158 . . . . . . . . 9 (𝑏 = 𝑐 → (2nd𝑏) = (2nd𝑐))
2019oveq2d 6631 . . . . . . . 8 (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑐)))
2118, 20oveq12d 6633 . . . . . . 7 (𝑏 = 𝑐 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
22 ovex 6643 . . . . . . 7 ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) ∈ V
2321, 2, 22fvmpt 6249 . . . . . 6 (𝑐 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
2423ad2antrl 763 . . . . 5 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
25 fveq2 6158 . . . . . . . 8 (𝑏 = 𝑑 → (1st𝑏) = (1st𝑑))
26 fveq2 6158 . . . . . . . . 9 (𝑏 = 𝑑 → (2nd𝑏) = (2nd𝑑))
2726oveq2d 6631 . . . . . . . 8 (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑑)))
2825, 27oveq12d 6633 . . . . . . 7 (𝑏 = 𝑑 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
29 ovex 6643 . . . . . . 7 ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ∈ V
3028, 2, 29fvmpt 6249 . . . . . 6 (𝑑 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
3130ad2antll 764 . . . . 5 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
3224, 31eqeq12d 2636 . . . 4 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) ↔ ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑)))))
33 rmspecsqrtnq 36989 . . . . . . . 8 (𝐴 ∈ (ℤ‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
3433adantr 481 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35 nn0ssq 11756 . . . . . . . 8 0 ⊆ ℚ
36 xp1st 7158 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (1st𝑐) ∈ ℕ0)
3736ad2antrl 763 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℕ0)
3835, 37sseldi 3586 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℚ)
39 xp2nd 7159 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (2nd𝑐) ∈ ℤ)
4039ad2antrl 763 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℤ)
41 zq 11754 . . . . . . . 8 ((2nd𝑐) ∈ ℤ → (2nd𝑐) ∈ ℚ)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℚ)
43 xp1st 7158 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (1st𝑑) ∈ ℕ0)
4443ad2antll 764 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℕ0)
4535, 44sseldi 3586 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℚ)
46 xp2nd 7159 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (2nd𝑑) ∈ ℤ)
4746ad2antll 764 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℤ)
48 zq 11754 . . . . . . . 8 ((2nd𝑑) ∈ ℤ → (2nd𝑑) ∈ ℚ)
4947, 48syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℚ)
50 qirropth 36992 . . . . . . 7 (((√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ) ∧ ((1st𝑐) ∈ ℚ ∧ (2nd𝑐) ∈ ℚ) ∧ ((1st𝑑) ∈ ℚ ∧ (2nd𝑑) ∈ ℚ)) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ↔ ((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑))))
5134, 38, 42, 45, 49, 50syl122anc 1332 . . . . . 6 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ↔ ((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑))))
5251biimpd 219 . . . . 5 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) → ((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑))))
53 xpopth 7167 . . . . . 6 ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5453adantl 482 . . . . 5 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5552, 54sylibd 229 . . . 4 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) → 𝑐 = 𝑑))
5632, 55sylbid 230 . . 3 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) → 𝑐 = 𝑑))
5756ralrimivva 2967 . 2 (𝐴 ∈ (ℤ‘2) → ∀𝑐 ∈ (ℕ0 × ℤ)∀𝑑 ∈ (ℕ0 × ℤ)(((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) → 𝑐 = 𝑑))
58 dff1o6 6496 . 2 ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ↔ ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) Fn (ℕ0 × ℤ) ∧ ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ∧ ∀𝑐 ∈ (ℕ0 × ℤ)∀𝑑 ∈ (ℕ0 × ℤ)(((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) → 𝑐 = 𝑑)))
594, 17, 57, 58syl3anbrc 1244 1 (𝐴 ∈ (ℤ‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2908  wrex 2909  cdif 3557  cop 4161  cmpt 4683   × cxp 5082  ran crn 5085   Fn wfn 5852  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  cc 9894  1c1 9897   + caddc 9899   · cmul 9901  cmin 10226  2c2 11030  0cn0 11252  cz 11337  cuz 11647  cq 11748  cexp 12816  csqrt 13923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-dvds 14927  df-gcd 15160  df-numer 15386  df-denom 15387
This theorem is referenced by:  rmxyelxp  36996  rmxyval  36999
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