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Theorem qdenval 16674
Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qdenval (𝐴 ∈ β„š β†’ (denomβ€˜π΄) = (2nd 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 qdenval
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2737 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯)) ↔ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))
21anbi2d 630 . . . 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 7367 . . 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 6896 . 2 (π‘Ž = 𝐴 β†’ (2nd β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))) = (2nd β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ 𝐴 = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))))
5 df-denom 16672 . 2 denom = (π‘Ž ∈ β„š ↦ (2nd β€˜(β„©π‘₯ ∈ (β„€ Γ— β„•)(((1st β€˜π‘₯) gcd (2nd β€˜π‘₯)) = 1 ∧ π‘Ž = ((1st β€˜π‘₯) / (2nd β€˜π‘₯))))))
6 fvex 6905 . 2 (2nd 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 6999 1 (𝐴 ∈ β„š β†’ (denomβ€˜π΄) = (2nd 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 397   = wceq 1542   ∈ wcel 2107   Γ— cxp 5675  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  1c1 11111   / cdiv 11871  β„•cn 12212  β„€cz 12558  β„šcq 12932   gcd cgcd 16435  denomcdenom 16670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-denom 16672
This theorem is referenced by:  qnumdencl  16675  fden  16679  qnumdenbi  16680
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