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Theorem rmxypairf1o 39392
Description: The function used to extract rational and irrational parts in df-rmx 39383 and df-rmy 39384 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 7183 . . . 4 ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ∈ V
2 eqid 2826 . . . 4 (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
31, 2fnmpti 6490 . . 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 3503 . . . . . . . . . 10 𝑐 ∈ V
6 vex 3503 . . . . . . . . . 10 𝑑 ∈ V
75, 6op1std 7695 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → (1st𝑏) = 𝑐)
85, 6op2ndd 7696 . . . . . . . . . 10 (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd𝑏) = 𝑑)
98oveq2d 7166 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
107, 9oveq12d 7168 . . . . . . . 8 (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1110eqeq2d 2837 . . . . . . 7 (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
1211rexxp 5712 . . . . . 6 (∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1312bicomi 225 . . . . 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 2890 . . 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 5826 . . 3 ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))}
1715, 16syl6reqr 2880 . 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 6669 . . . . . . . 8 (𝑏 = 𝑐 → (1st𝑏) = (1st𝑐))
19 fveq2 6669 . . . . . . . . 9 (𝑏 = 𝑐 → (2nd𝑏) = (2nd𝑐))
2019oveq2d 7166 . . . . . . . 8 (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑐)))
2118, 20oveq12d 7168 . . . . . . 7 (𝑏 = 𝑐 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
22 ovex 7183 . . . . . . 7 ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) ∈ V
2321, 2, 22fvmpt 6767 . . . . . 6 (𝑐 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
2423ad2antrl 724 . . . . 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 6669 . . . . . . . 8 (𝑏 = 𝑑 → (1st𝑏) = (1st𝑑))
26 fveq2 6669 . . . . . . . . 9 (𝑏 = 𝑑 → (2nd𝑏) = (2nd𝑑))
2726oveq2d 7166 . . . . . . . 8 (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑑)))
2825, 27oveq12d 7168 . . . . . . 7 (𝑏 = 𝑑 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
29 ovex 7183 . . . . . . 7 ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ∈ V
3028, 2, 29fvmpt 6767 . . . . . 6 (𝑑 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
3130ad2antll 725 . . . . 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 2842 . . . 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 39387 . . . . . . . 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 12351 . . . . . . . 8 0 ⊆ ℚ
36 xp1st 7717 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (1st𝑐) ∈ ℕ0)
3736ad2antrl 724 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℕ0)
3835, 37sseldi 3969 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℚ)
39 xp2nd 7718 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (2nd𝑐) ∈ ℤ)
4039ad2antrl 724 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℤ)
41 zq 12348 . . . . . . . 8 ((2nd𝑐) ∈ ℤ → (2nd𝑐) ∈ ℚ)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℚ)
43 xp1st 7717 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (1st𝑑) ∈ ℕ0)
4443ad2antll 725 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℕ0)
4535, 44sseldi 3969 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℚ)
46 xp2nd 7718 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (2nd𝑑) ∈ ℤ)
4746ad2antll 725 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℤ)
48 zq 12348 . . . . . . . 8 ((2nd𝑑) ∈ ℤ → (2nd𝑑) ∈ ℚ)
4947, 48syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℚ)
50 qirropth 39389 . . . . . . 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 1373 . . . . . 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 230 . . . . 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 7726 . . . . . 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 240 . . . 4 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) → 𝑐 = 𝑑))
5632, 55sylbid 241 . . 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 3196 . 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 7028 . 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 1337 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 207  wa 396   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  cdif 3937  cop 4570  cmpt 5143   × cxp 5552  ran crn 5555   Fn wfn 6349  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  cc 10529  1c1 10532   + caddc 10534   · cmul 10536  cmin 10864  2c2 11686  0cn0 11891  cz 11975  cuz 12237  cq 12342  cexp 13424  csqrt 14587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12385  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13425  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-dvds 15603  df-gcd 15839  df-numer 16070  df-denom 16071
This theorem is referenced by:  rmxyelxp  39393  rmxyval  39396
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