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Theorem qnumval 11790
Description: Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumval (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem qnumval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2124 . . . . 5 (𝑎 = 𝐴 → (𝑎 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑥) / (2nd𝑥))))
21anbi2d 459 . . . 4 (𝑎 = 𝐴 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
32riotabidv 5700 . . 3 (𝑎 = 𝐴 → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥)))) = (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
43fveq2d 5393 . 2 (𝑎 = 𝐴 → (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))) = (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
5 df-numer 11788 . 2 numer = (𝑎 ∈ ℚ ↦ (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))))
6 zex 9031 . . . 4 ℤ ∈ V
7 nnex 8694 . . . 4 ℕ ∈ V
86, 7xpex 4624 . . 3 (ℤ × ℕ) ∈ V
9 riotaexg 5702 . . 3 ((ℤ × ℕ) ∈ V → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))) ∈ V)
10 1stexg 6033 . . 3 ((𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))) ∈ V → (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))) ∈ V)
118, 9, 10mp2b 8 . 2 (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))) ∈ V
124, 5, 11fvmpt 5466 1 (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660   × cxp 4507  cfv 5093  crio 5697  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  1c1 7589   / cdiv 8400  cn 8688  cz 9022  cq 9379   gcd cgcd 11562  numercnumer 11786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-riota 5698  df-ov 5745  df-1st 6006  df-neg 7904  df-inn 8689  df-z 9023  df-numer 11788
This theorem is referenced by:  qnumdencl  11792  fnum  11795  qnumdenbi  11797
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