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Theorem qdenval 12760
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 2238 . . . . 5 (𝑎 = 𝐴 → (𝑎 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑥) / (2nd𝑥))))
21anbi2d 464 . . . 4 (𝑎 = 𝐴 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
32riotabidv 5973 . . 3 (𝑎 = 𝐴 → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥)))) = (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
43fveq2d 5643 . 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 12758 . 2 denom = (𝑎 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))))
6 zex 9488 . . . 4 ℤ ∈ V
7 nnex 9149 . . . 4 ℕ ∈ V
86, 7xpex 4842 . . 3 (ℤ × ℕ) ∈ V
9 riotaexg 5975 . . 3 ((ℤ × ℕ) ∈ V → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))) ∈ V)
10 2ndexg 6331 . . 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 5723 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 104   = wceq 1397  wcel 2202  Vcvv 2802   × cxp 4723  cfv 5326  crio 5970  (class class class)co 6018  1st c1st 6301  2nd c2nd 6302  1c1 8033   / cdiv 8852  cn 9143  cz 9479  cq 9853   gcd cgcd 12526  denomcdenom 12756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129
This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-2nd 6304  df-neg 8353  df-inn 9144  df-z 9480  df-denom 12758
This theorem is referenced by:  qnumdencl  12761  fden  12765  qnumdenbi  12766
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