Theorem qdenval 12508

Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Assertion

Ref	Expression
qdenval	|- ( A e. QQ -> (denom ` A ) = ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )

Distinct variable group:   x, A

Proof of Theorem qdenval

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2212	. . . . 5 |- ( a = A -> ( a = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )
2	1	anbi2d 464	. . . 4 |- ( a = A -> ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ a = ( ( 1st ` x ) / ( 2nd ` x ) ) ) <-> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) )
3	2	riotabidv 5901	. . 3 |- ( a = A -> ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ a = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) = ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) )
4	3	fveq2d 5580	. 2 |- ( a = A -> ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ a = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) = ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
5		df-denom 12506	. 2 |- denom = ( a e. QQ |-> ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ a = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
6		zex 9381	. . . 4 |- ZZ e. _V
7		nnex 9042	. . . 4 |- NN e. _V
8	6, 7	xpex 4790	. . 3 |- ( ZZ X. NN ) e. _V
9		riotaexg 5903	. . 3 |- ( ( ZZ X. NN ) e. _V -> ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) e. _V )
10		2ndexg 6254	. . 3 |- ( ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) e. _V -> ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) e. _V )
11	8, 9, 10	mp2b 8	. 2 |- ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) e. _V
12	4, 5, 11	fvmpt 5656	1 |- ( A e. QQ -> (denom ` A ) = ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   X. cxp 4673   ` cfv 5271   iota_crio 5898  (class class class)co 5944   1stc1st 6224   2ndc2nd 6225   1c1 7926   / cdiv 8745   NNcn 9036   ZZcz 9372   QQcq 9740   gcd cgcd 12274  denomcdenom 12504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-riota 5899  df-ov 5947  df-2nd 6227  df-neg 8246  df-inn 9037  df-z 9373  df-denom 12506

This theorem is referenced by:  qnumdencl  12509  fden  12513  qnumdenbi  12514