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Theorem qdenval 11709
 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 2121 . . . . 5 (𝑎 = 𝐴 → (𝑎 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑥) / (2nd𝑥))))
21anbi2d 457 . . . 4 (𝑎 = 𝐴 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
32riotabidv 5686 . . 3 (𝑎 = 𝐴 → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥)))) = (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
43fveq2d 5379 . 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 11707 . 2 denom = (𝑎 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))))
6 zex 8967 . . . 4 ℤ ∈ V
7 nnex 8636 . . . 4 ℕ ∈ V
86, 7xpex 4614 . . 3 (ℤ × ℕ) ∈ V
9 riotaexg 5688 . . 3 ((ℤ × ℕ) ∈ V → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))) ∈ V)
10 2ndexg 6020 . . 3 ((𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))) ∈ V → (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))) ∈ V)
118, 9, 10mp2b 8 . 2 (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))) ∈ V
124, 5, 11fvmpt 5452 1 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1314   ∈ wcel 1463  Vcvv 2657   × cxp 4497  ‘cfv 5081  ℩crio 5683  (class class class)co 5728  1st c1st 5990  2nd c2nd 5991  1c1 7548   / cdiv 8345  ℕcn 8630  ℤcz 8958  ℚcq 9313   gcd cgcd 11483  denomcdenom 11705 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-cnex 7636  ax-resscn 7637  ax-1re 7639  ax-addrcl 7642 This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fo 5087  df-fv 5089  df-riota 5684  df-ov 5731  df-2nd 5993  df-neg 7859  df-inn 8631  df-z 8959  df-denom 11707 This theorem is referenced by:  qnumdencl  11710  fden  11714  qnumdenbi  11715
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