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Theorem rmxypairf1o 42899
Description: The function used to extract rational and irrational parts in df-rmx 42889 and df-rmy 42890 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 7463 . . . 4 ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ∈ V
2 eqid 2734 . . . 4 (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
31, 2fnmpti 6711 . . 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))
52rnmpt 5970 . . 3 ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))}
6 vex 3481 . . . . . . . . . 10 𝑐 ∈ V
7 vex 3481 . . . . . . . . . 10 𝑑 ∈ V
86, 7op1std 8022 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → (1st𝑏) = 𝑐)
96, 7op2ndd 8023 . . . . . . . . . 10 (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd𝑏) = 𝑑)
109oveq2d 7446 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
118, 10oveq12d 7448 . . . . . . . 8 (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1211eqeq2d 2745 . . . . . . 7 (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
1312rexxp 5855 . . . . . 6 (∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1413bicomi 224 . . . . 5 (∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
1514a1i 11 . . . 4 (𝐴 ∈ (ℤ‘2) → (∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ↔ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
1615abbidv 2805 . . 3 (𝐴 ∈ (ℤ‘2) → {𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))})
175, 16eqtr4id 2793 . 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 6906 . . . . . . . 8 (𝑏 = 𝑐 → (1st𝑏) = (1st𝑐))
19 fveq2 6906 . . . . . . . . 9 (𝑏 = 𝑐 → (2nd𝑏) = (2nd𝑐))
2019oveq2d 7446 . . . . . . . 8 (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑐)))
2118, 20oveq12d 7448 . . . . . . 7 (𝑏 = 𝑐 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
22 ovex 7463 . . . . . . 7 ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) ∈ V
2321, 2, 22fvmpt 7015 . . . . . 6 (𝑐 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
2423ad2antrl 728 . . . . 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 6906 . . . . . . . 8 (𝑏 = 𝑑 → (1st𝑏) = (1st𝑑))
26 fveq2 6906 . . . . . . . . 9 (𝑏 = 𝑑 → (2nd𝑏) = (2nd𝑑))
2726oveq2d 7446 . . . . . . . 8 (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑑)))
2825, 27oveq12d 7448 . . . . . . 7 (𝑏 = 𝑑 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
29 ovex 7463 . . . . . . 7 ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ∈ V
3028, 2, 29fvmpt 7015 . . . . . 6 (𝑑 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
3130ad2antll 729 . . . . 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 2750 . . . 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 42893 . . . . . . . 8 (𝐴 ∈ (ℤ‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
3433adantr 480 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35 nn0ssq 12996 . . . . . . . 8 0 ⊆ ℚ
36 xp1st 8044 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (1st𝑐) ∈ ℕ0)
3736ad2antrl 728 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℕ0)
3835, 37sselid 3992 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℚ)
39 xp2nd 8045 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (2nd𝑐) ∈ ℤ)
4039ad2antrl 728 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℤ)
41 zq 12993 . . . . . . . 8 ((2nd𝑐) ∈ ℤ → (2nd𝑐) ∈ ℚ)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℚ)
43 xp1st 8044 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (1st𝑑) ∈ ℕ0)
4443ad2antll 729 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℕ0)
4535, 44sselid 3992 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℚ)
46 xp2nd 8045 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (2nd𝑑) ∈ ℤ)
4746ad2antll 729 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℤ)
48 zq 12993 . . . . . . . 8 ((2nd𝑑) ∈ ℤ → (2nd𝑑) ∈ ℚ)
4947, 48syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℚ)
50 qirropth 42895 . . . . . . 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 1378 . . . . . 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 229 . . . . 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 8053 . . . . . 6 ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5453adantl 481 . . . . 5 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5552, 54sylibd 239 . . . 4 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) → 𝑐 = 𝑑))
5632, 55sylbid 240 . . 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 3199 . 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 7294 . 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 1342 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 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  cdif 3959  cop 4636  cmpt 5230   × cxp 5686  ran crn 5689   Fn wfn 6557  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  cc 11150  1c1 11153   + caddc 11155   · cmul 11157  cmin 11489  2c2 12318  0cn0 12523  cz 12610  cuz 12875  cq 12987  cexp 14098  csqrt 15268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-gcd 16528  df-numer 16768  df-denom 16769
This theorem is referenced by:  rmxyelxp  42900  rmxyval  42903
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