Theorem qnumval 12747

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 2236	. . . . 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 5968	. . 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 5639	. 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 12745	. 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 9478	. . . 4 |- ZZ e. _V
7		nnex 9139	. . . 4 |- NN e. _V
8	6, 7	xpex 4840	. . 3 |- ( ZZ X. NN ) e. _V
9		riotaexg 5970	. . 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 6325	. . 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 5719	1 |- ( A e. QQ -> (numer ` A ) = ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   X. cxp 4721   ` cfv 5324   iota_crio 5965  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   1c1 8023   / cdiv 8842   NNcn 9133   ZZcz 9469   QQcq 9843   gcd cgcd 12514  numercnumer 12743

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-1st 6298  df-neg 8343  df-inn 9134  df-z 9470  df-numer 12745

This theorem is referenced by:  qnumdencl  12749  fnum  12752  qnumdenbi  12754