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Theorem rmxypairf1o 37947
Description: The function used to extract rational and irrational parts in df-rmx 37937 and df-rmy 37938 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 6829 . . . 4 ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ∈ V
2 eqid 2748 . . . 4 (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
31, 2fnmpti 6171 . . 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 3331 . . . . . . . . . 10 𝑐 ∈ V
6 vex 3331 . . . . . . . . . 10 𝑑 ∈ V
75, 6op1std 7331 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → (1st𝑏) = 𝑐)
85, 6op2ndd 7332 . . . . . . . . . 10 (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd𝑏) = 𝑑)
98oveq2d 6817 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
107, 9oveq12d 6819 . . . . . . . 8 (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1110eqeq2d 2758 . . . . . . 7 (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
1211rexxp 5408 . . . . . 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 2867 . . 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 5514 . . 3 ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))}
1715, 16syl6reqr 2801 . 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 6340 . . . . . . . 8 (𝑏 = 𝑐 → (1st𝑏) = (1st𝑐))
19 fveq2 6340 . . . . . . . . 9 (𝑏 = 𝑐 → (2nd𝑏) = (2nd𝑐))
2019oveq2d 6817 . . . . . . . 8 (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑐)))
2118, 20oveq12d 6819 . . . . . . 7 (𝑏 = 𝑐 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
22 ovex 6829 . . . . . . 7 ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) ∈ V
2321, 2, 22fvmpt 6432 . . . . . 6 (𝑐 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
2423ad2antrl 766 . . . . 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 6340 . . . . . . . 8 (𝑏 = 𝑑 → (1st𝑏) = (1st𝑑))
26 fveq2 6340 . . . . . . . . 9 (𝑏 = 𝑑 → (2nd𝑏) = (2nd𝑑))
2726oveq2d 6817 . . . . . . . 8 (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑑)))
2825, 27oveq12d 6819 . . . . . . 7 (𝑏 = 𝑑 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
29 ovex 6829 . . . . . . 7 ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ∈ V
3028, 2, 29fvmpt 6432 . . . . . 6 (𝑑 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
3130ad2antll 767 . . . . 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 2763 . . . 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 37941 . . . . . . . 8 (𝐴 ∈ (ℤ‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
3433adantr 472 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35 nn0ssq 11960 . . . . . . . 8 0 ⊆ ℚ
36 xp1st 7353 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (1st𝑐) ∈ ℕ0)
3736ad2antrl 766 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℕ0)
3835, 37sseldi 3730 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℚ)
39 xp2nd 7354 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (2nd𝑐) ∈ ℤ)
4039ad2antrl 766 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℤ)
41 zq 11958 . . . . . . . 8 ((2nd𝑐) ∈ ℤ → (2nd𝑐) ∈ ℚ)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℚ)
43 xp1st 7353 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (1st𝑑) ∈ ℕ0)
4443ad2antll 767 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℕ0)
4535, 44sseldi 3730 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℚ)
46 xp2nd 7354 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (2nd𝑑) ∈ ℤ)
4746ad2antll 767 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℤ)
48 zq 11958 . . . . . . . 8 ((2nd𝑑) ∈ ℤ → (2nd𝑑) ∈ ℚ)
4947, 48syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℚ)
50 qirropth 37944 . . . . . . 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 1472 . . . . . 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 7362 . . . . . 6 ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5453adantl 473 . . . . 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 3097 . 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 6682 . 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 1407 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 383   = wceq 1620  wcel 2127  {cab 2734  wral 3038  wrex 3039  cdif 3700  cop 4315  cmpt 4869   × cxp 5252  ran crn 5255   Fn wfn 6032  1-1-ontowf1o 6036  cfv 6037  (class class class)co 6801  1st c1st 7319  2nd c2nd 7320  cc 10097  1c1 10100   + caddc 10102   · cmul 10104  cmin 10429  2c2 11233  0cn0 11455  cz 11540  cuz 11850  cq 11952  cexp 13025  csqrt 14143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8501  df-inf 8502  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-n0 11456  df-z 11541  df-uz 11851  df-q 11953  df-rp 11997  df-fl 12758  df-mod 12834  df-seq 12967  df-exp 13026  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-dvds 15154  df-gcd 15390  df-numer 15616  df-denom 15617
This theorem is referenced by:  rmxyelxp  37948  rmxyval  37951
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