Theorem recmulnqg 7192

Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)

Assertion

Ref	Expression
recmulnqg	|- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )

Proof of Theorem recmulnqg

Dummy variables x y z w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5774	. . . . 5 |- ( x = A -> ( x .Q y ) = ( A .Q y ) )
2	1	eqeq1d 2146	. . . 4 |- ( x = A -> ( ( x .Q y ) = 1Q <-> ( A .Q y ) = 1Q ) )
3	2	anbi2d 459	. . 3 |- ( x = A -> ( ( y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( y e. Q. /\ ( A .Q y ) = 1Q ) ) )
4		eleq1 2200	. . . 4 |- ( y = B -> ( y e. Q. <-> B e. Q. ) )
5		oveq2 5775	. . . . 5 |- ( y = B -> ( A .Q y ) = ( A .Q B ) )
6	5	eqeq1d 2146	. . . 4 |- ( y = B -> ( ( A .Q y ) = 1Q <-> ( A .Q B ) = 1Q ) )
7	4, 6	anbi12d 464	. . 3 |- ( y = B -> ( ( y e. Q. /\ ( A .Q y ) = 1Q ) <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
8		recexnq 7191	. . . 4 |- ( x e. Q. -> E. y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
9		1nq 7167	. . . . 5 |- 1Q e. Q.
10		mulcomnqg 7184	. . . . 5 |- ( ( z e. Q. /\ w e. Q. ) -> ( z .Q w ) = ( w .Q z ) )
11		mulassnqg 7185	. . . . 5 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( ( z .Q w ) .Q v ) = ( z .Q ( w .Q v ) ) )
12		mulidnq 7190	. . . . 5 |- ( z e. Q. -> ( z .Q 1Q ) = z )
13	9, 10, 11, 12	caovimo 5957	. . . 4 |- ( x e. Q. -> E* y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
14		eu5 2044	. . . 4 |- ( E! y ( y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( E. y ( y e. Q. /\ ( x .Q y ) = 1Q ) /\ E* y ( y e. Q. /\ ( x .Q y ) = 1Q ) ) )
15	8, 13, 14	sylanbrc 413	. . 3 |- ( x e. Q. -> E! y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
16		df-rq 7153	. . . 4 |- *Q = { <. x , y >. | ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) }
17		3anass 966	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) )
18	17	opabbii 3990	. . . 4 |- { <. x , y >. | ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) } = { <. x , y >. | ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) }
19	16, 18	eqtri 2158	. . 3 |- *Q = { <. x , y >. | ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) }
20	3, 7, 15, 19	fvopab3g 5487	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
21		ibar 299	. . 3 |- ( B e. Q. -> ( ( A .Q B ) = 1Q <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
22	21	adantl 275	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( A .Q B ) = 1Q <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
23	20, 22	bitr4d 190	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   E!weu 1997   E*wmo 1998   {copab 3983   ` cfv 5118  (class class class)co 5767   Q.cnq 7081   1Qc1q 7082   .Q cmq 7084   *Qcrq 7085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152  df-rq 7153

This theorem is referenced by:  recclnq  7193  recidnq  7194  recrecnq  7195  recexprlem1ssl  7434  recexprlem1ssu  7435