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Theorem qnumval 16716
Description: Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumval (𝐴 ∈ β„š β†’ (numerβ€˜π΄) = (1st β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem qnumval
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2732 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯)) ↔ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))
21anbi2d 628 . . . 4 (π‘Ž = 𝐴 β†’ ((((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))) ↔ (((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯)))))
32riotabidv 7384 . . 3 (π‘Ž = 𝐴 β†’ (β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯)))) = (β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯)))))
43fveq2d 6906 . 2 (π‘Ž = 𝐴 β†’ (1st β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))) = (1st β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))))
5 df-numer 16714 . 2 numer = (π‘Ž ∈ β„š ↦ (1st β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))))
6 fvex 6915 . 2 (1st β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))) ∈ V
74, 5, 6fvmpt 7010 1 (𝐴 ∈ β„š β†’ (numerβ€˜π΄) = (1st β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   Γ— cxp 5680  β€˜cfv 6553  β„©crio 7381  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  1c1 11147   / cdiv 11909  β„•cn 12250  β„€cz 12596  β„šcq 12970   gcd cgcd 16476  numercnumer 16712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382  df-numer 16714
This theorem is referenced by:  qnumdencl  16718  fnum  16721  qnumdenbi  16723
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