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Theorem qdenval 15284
 Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qdenval (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem qdenval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2614 . . . . 5 (𝑎 = 𝐴 → (𝑎 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑥) / (2nd𝑥))))
21anbi2d 736 . . . 4 (𝑎 = 𝐴 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
32riotabidv 6513 . . 3 (𝑎 = 𝐴 → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥)))) = (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
43fveq2d 6107 . 2 (𝑎 = 𝐴 → (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))) = (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
5 df-denom 15282 . 2 denom = (𝑎 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))))
6 fvex 6113 . 2 (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))) ∈ V
74, 5, 6fvmpt 6191 1 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   × cxp 5036  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1c1 9816   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℚcq 11664   gcd cgcd 15054  denomcdenom 15280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-denom 15282 This theorem is referenced by:  qnumdencl  15285  fden  15289  qnumdenbi  15290
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