Theorem qdenval 12188

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 2184	. . . . 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 5835	. . 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 5521	. 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 12186	. 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 9264	. . . 4 |- ZZ e. _V
7		nnex 8927	. . . 4 |- NN e. _V
8	6, 7	xpex 4743	. . 3 |- ( ZZ X. NN ) e. _V
9		riotaexg 5837	. . 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 6171	. . 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 5595	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 1353   e. wcel 2148   _Vcvv 2739   X. cxp 4626   ` cfv 5218   iota_crio 5832  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   1c1 7814   / cdiv 8631   NNcn 8921   ZZcz 9255   QQcq 9621   gcd cgcd 11945  denomcdenom 12184

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226  df-riota 5833  df-ov 5880  df-2nd 6144  df-neg 8133  df-inn 8922  df-z 9256  df-denom 12186

This theorem is referenced by:  qnumdencl  12189  fden  12193  qnumdenbi  12194