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Theorem rmxypairf1o 38261
Description: The function used to extract rational and irrational parts in df-rmx 38252 and df-rmy 38253 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 6910 . . . 4 ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ∈ V
2 eqid 2799 . . . 4 (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
31, 2fnmpti 6233 . . 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 3388 . . . . . . . . . 10 𝑐 ∈ V
6 vex 3388 . . . . . . . . . 10 𝑑 ∈ V
75, 6op1std 7411 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → (1st𝑏) = 𝑐)
85, 6op2ndd 7412 . . . . . . . . . 10 (𝑏 = ⟨𝑐, 𝑑⟩ → (2nd𝑏) = 𝑑)
98oveq2d 6894 . . . . . . . . 9 (𝑏 = ⟨𝑐, 𝑑⟩ → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · 𝑑))
107, 9oveq12d 6896 . . . . . . . 8 (𝑏 = ⟨𝑐, 𝑑⟩ → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1110eqeq2d 2809 . . . . . . 7 (𝑏 = ⟨𝑐, 𝑑⟩ → (𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))))
1211rexxp 5468 . . . . . 6 (∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) ↔ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
1312bicomi 216 . . . . 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 2918 . . 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 5575 . . 3 ran (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))) = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 × ℤ)𝑎 = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))}
1715, 16syl6reqr 2852 . 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 6411 . . . . . . . 8 (𝑏 = 𝑐 → (1st𝑏) = (1st𝑐))
19 fveq2 6411 . . . . . . . . 9 (𝑏 = 𝑐 → (2nd𝑏) = (2nd𝑐))
2019oveq2d 6894 . . . . . . . 8 (𝑏 = 𝑐 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑐)))
2118, 20oveq12d 6896 . . . . . . 7 (𝑏 = 𝑐 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
22 ovex 6910 . . . . . . 7 ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) ∈ V
2321, 2, 22fvmpt 6507 . . . . . 6 (𝑐 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑐) = ((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))))
2423ad2antrl 720 . . . . 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 6411 . . . . . . . 8 (𝑏 = 𝑑 → (1st𝑏) = (1st𝑑))
26 fveq2 6411 . . . . . . . . 9 (𝑏 = 𝑑 → (2nd𝑏) = (2nd𝑑))
2726oveq2d 6894 . . . . . . . 8 (𝑏 = 𝑑 → ((√‘((𝐴↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑑)))
2825, 27oveq12d 6896 . . . . . . 7 (𝑏 = 𝑑 → ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
29 ovex 6910 . . . . . . 7 ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) ∈ V
3028, 2, 29fvmpt 6507 . . . . . 6 (𝑑 ∈ (ℕ0 × ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘𝑑) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))))
3130ad2antll 721 . . . . 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 2814 . . . 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 38256 . . . . . . . 8 (𝐴 ∈ (ℤ‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
3433adantr 473 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
35 nn0ssq 12041 . . . . . . . 8 0 ⊆ ℚ
36 xp1st 7433 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (1st𝑐) ∈ ℕ0)
3736ad2antrl 720 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℕ0)
3835, 37sseldi 3796 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑐) ∈ ℚ)
39 xp2nd 7434 . . . . . . . . 9 (𝑐 ∈ (ℕ0 × ℤ) → (2nd𝑐) ∈ ℤ)
4039ad2antrl 720 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℤ)
41 zq 12039 . . . . . . . 8 ((2nd𝑐) ∈ ℤ → (2nd𝑐) ∈ ℚ)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑐) ∈ ℚ)
43 xp1st 7433 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (1st𝑑) ∈ ℕ0)
4443ad2antll 721 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℕ0)
4535, 44sseldi 3796 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (1st𝑑) ∈ ℚ)
46 xp2nd 7434 . . . . . . . . 9 (𝑑 ∈ (ℕ0 × ℤ) → (2nd𝑑) ∈ ℤ)
4746ad2antll 721 . . . . . . . 8 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℤ)
48 zq 12039 . . . . . . . 8 ((2nd𝑑) ∈ ℤ → (2nd𝑑) ∈ ℚ)
4947, 48syl 17 . . . . . . 7 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (2nd𝑑) ∈ ℚ)
50 qirropth 38258 . . . . . . 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 1499 . . . . . 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 221 . . . . 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 7442 . . . . . 6 ((𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ)) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5453adantl 474 . . . . 5 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) = (1st𝑑) ∧ (2nd𝑐) = (2nd𝑑)) ↔ 𝑐 = 𝑑))
5552, 54sylibd 231 . . . 4 ((𝐴 ∈ (ℤ‘2) ∧ (𝑐 ∈ (ℕ0 × ℤ) ∧ 𝑑 ∈ (ℕ0 × ℤ))) → (((1st𝑐) + ((√‘((𝐴↑2) − 1)) · (2nd𝑐))) = ((1st𝑑) + ((√‘((𝐴↑2) − 1)) · (2nd𝑑))) → 𝑐 = 𝑑))
5632, 55sylbid 232 . . 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 3152 . 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 6759 . 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 1444 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 198  wa 385   = wceq 1653  wcel 2157  {cab 2785  wral 3089  wrex 3090  cdif 3766  cop 4374  cmpt 4922   × cxp 5310  ran crn 5313   Fn wfn 6096  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  cc 10222  1c1 10225   + caddc 10227   · cmul 10229  cmin 10556  2c2 11368  0cn0 11580  cz 11666  cuz 11930  cq 12033  cexp 13114  csqrt 14314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-z 11667  df-uz 11931  df-q 12034  df-rp 12075  df-fl 12848  df-mod 12924  df-seq 13056  df-exp 13115  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-abs 14317  df-dvds 15320  df-gcd 15552  df-numer 15776  df-denom 15777
This theorem is referenced by:  rmxyelxp  38262  rmxyval  38265
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