Theorem recmulnqg 7722

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 6065	. . . . 5 |- ( x = A -> ( x .Q y ) = ( A .Q y ) )
2	1	eqeq1d 2243	. . . 4 |- ( x = A -> ( ( x .Q y ) = 1Q <-> ( A .Q y ) = 1Q ) )
3	2	anbi2d 464	. . 3 |- ( x = A -> ( ( y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( y e. Q. /\ ( A .Q y ) = 1Q ) ) )
4		eleq1 2297	. . . 4 |- ( y = B -> ( y e. Q. <-> B e. Q. ) )
5		oveq2 6066	. . . . 5 |- ( y = B -> ( A .Q y ) = ( A .Q B ) )
6	5	eqeq1d 2243	. . . 4 |- ( y = B -> ( ( A .Q y ) = 1Q <-> ( A .Q B ) = 1Q ) )
7	4, 6	anbi12d 473	. . 3 |- ( y = B -> ( ( y e. Q. /\ ( A .Q y ) = 1Q ) <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
8		recexnq 7721	. . . 4 |- ( x e. Q. -> E. y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
9		1nq 7697	. . . . 5 |- 1Q e. Q.
10		mulcomnqg 7714	. . . . 5 |- ( ( z e. Q. /\ w e. Q. ) -> ( z .Q w ) = ( w .Q z ) )
11		mulassnqg 7715	. . . . 5 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( ( z .Q w ) .Q v ) = ( z .Q ( w .Q v ) ) )
12		mulidnq 7720	. . . . 5 |- ( z e. Q. -> ( z .Q 1Q ) = z )
13	9, 10, 11, 12	caovimo 6256	. . . 4 |- ( x e. Q. -> E* y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
14		eu5 2130	. . . 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 417	. . 3 |- ( x e. Q. -> E! y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
16		df-rq 7683	. . . 4 |- *Q = { <. x , y >. | ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) }
17		3anass 1009	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) )
18	17	opabbii 4182	. . . 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 2255	. . 3 |- *Q = { <. x , y >. | ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) }
20	3, 7, 15, 19	fvopab3g 5755	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
21		ibar 301	. . 3 |- ( B e. Q. -> ( ( A .Q B ) = 1Q <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
22	21	adantl 277	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( A .Q B ) = 1Q <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
23	20, 22	bitr4d 191	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   E!weu 2082   E*wmo 2083   e. wcel 2205   {copab 4175   ` cfv 5357  (class class class)co 6058   Q.cnq 7611   1Qc1q 7612   .Q cmq 7614   *Qcrq 7615

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-mi 7637  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-mqqs 7681  df-1nqqs 7682  df-rq 7683

This theorem is referenced by:  recclnq  7723  recidnq  7724  recrecnq  7725  recexprlem1ssl  7964  recexprlem1ssu  7965