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Theorem qdenval 16620
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 2741 . . . . 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 7320 . . 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 6851 . 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 16618 . 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 6860 . 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 6953 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 5636  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  1c1 11059   / cdiv 11819  β„•cn 12160  β„€cz 12506  β„šcq 12880   gcd cgcd 16381  denomcdenom 16616
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-denom 16618
This theorem is referenced by:  qnumdencl  16621  fden  16625  qnumdenbi  16626
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