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Theorem qnumval 11038
 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 2091 . . . . 5 (𝑎 = 𝐴 → (𝑎 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑥) / (2nd𝑥))))
21anbi2d 452 . . . 4 (𝑎 = 𝐴 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
32riotabidv 5571 . . 3 (𝑎 = 𝐴 → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥)))) = (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))))
43fveq2d 5272 . 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 11036 . 2 numer = (𝑎 ∈ ℚ ↦ (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑎 = ((1st𝑥) / (2nd𝑥))))))
6 zex 8692 . . . 4 ℤ ∈ V
7 nnex 8363 . . . 4 ℕ ∈ V
86, 7xpex 4521 . . 3 (ℤ × ℕ) ∈ V
9 riotaexg 5573 . . 3 ((ℤ × ℕ) ∈ V → (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥)))) ∈ V)
10 1stexg 5895 . . 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 5344 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 102   = wceq 1287   ∈ wcel 1436  Vcvv 2615   × cxp 4409  ‘cfv 4981  ℩crio 5568  (class class class)co 5613  1st c1st 5866  2nd c2nd 5867  1c1 7295   / cdiv 8078  ℕcn 8357  ℤcz 8683  ℚcq 9036   gcd cgcd 10813  numercnumer 11034 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-cnex 7380  ax-resscn 7381  ax-1re 7383  ax-addrcl 7386 This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fo 4987  df-fv 4989  df-riota 5569  df-ov 5616  df-1st 5868  df-neg 7600  df-inn 8358  df-z 8684  df-numer 11036 This theorem is referenced by:  qnumdencl  11040  fnum  11043  qnumdenbi  11045
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