Theorem qnumval 12990

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

Assertion

Ref	Expression
qnumval	|- ( A e. QQ -> (numer ` A ) = ( 1st ` ( 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 qnumval

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2245	. . . . 5 |- ( a = A -> ( a = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )
2	1	anbi2d 468	. . . 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 6043	. . 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 5702	. 2 |- ( a = A -> ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ a = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) = ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
5		df-numer 12988	. 2 |- numer = ( a e. QQ |-> ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ a = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
6		zex 9664	. . . 4 |- ZZ e. _V
7		nnex 9319	. . . 4 |- NN e. _V
8	6, 7	xpex 4893	. . 3 |- ( ZZ X. NN ) e. _V
9		riotaexg 6045	. . 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		1stexg 6405	. . 3 |- ( ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) e. _V -> ( 1st ` ( 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 |- ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) e. _V
12	4, 5, 11	fvmpt 5785	1 |- ( A e. QQ -> (numer ` A ) = ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   X. cxp 4774   ` cfv 5380   iota_crio 6040  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   1c1 8186   / cdiv 9010   NNcn 9313   ZZcz 9655   QQcq 10035   gcd cgcd 12755  numercnumer 12986

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-riota 6041  df-ov 6091  df-1st 6378  df-neg 8507  df-inn 9314  df-z 9656  df-numer 12988

This theorem is used by:  qnumdencl  12992  fnum  12995  qnumdenbi  12997