Theorem recmulnqg 6950

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 5659	. . . . 5 |- ( x = A -> ( x .Q y ) = ( A .Q y ) )
2	1	eqeq1d 2096	. . . 4 |- ( x = A -> ( ( x .Q y ) = 1Q <-> ( A .Q y ) = 1Q ) )
3	2	anbi2d 452	. . 3 |- ( x = A -> ( ( y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( y e. Q. /\ ( A .Q y ) = 1Q ) ) )
4		eleq1 2150	. . . 4 |- ( y = B -> ( y e. Q. <-> B e. Q. ) )
5		oveq2 5660	. . . . 5 |- ( y = B -> ( A .Q y ) = ( A .Q B ) )
6	5	eqeq1d 2096	. . . 4 |- ( y = B -> ( ( A .Q y ) = 1Q <-> ( A .Q B ) = 1Q ) )
7	4, 6	anbi12d 457	. . 3 |- ( y = B -> ( ( y e. Q. /\ ( A .Q y ) = 1Q ) <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
8		recexnq 6949	. . . 4 |- ( x e. Q. -> E. y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
9		1nq 6925	. . . . 5 |- 1Q e. Q.
10		mulcomnqg 6942	. . . . 5 |- ( ( z e. Q. /\ w e. Q. ) -> ( z .Q w ) = ( w .Q z ) )
11		mulassnqg 6943	. . . . 5 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( ( z .Q w ) .Q v ) = ( z .Q ( w .Q v ) ) )
12		mulidnq 6948	. . . . 5 |- ( z e. Q. -> ( z .Q 1Q ) = z )
13	9, 10, 11, 12	caovimo 5838	. . . 4 |- ( x e. Q. -> E* y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
14		eu5 1995	. . . 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 408	. . 3 |- ( x e. Q. -> E! y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
16		df-rq 6911	. . . 4 |- *Q = { <. x , y >. | ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) }
17		3anass 928	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) )
18	17	opabbii 3905	. . . 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 2108	. . 3 |- *Q = { <. x , y >. | ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) }
20	3, 7, 15, 19	fvopab3g 5377	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
21		ibar 295	. . 3 |- ( B e. Q. -> ( ( A .Q B ) = 1Q <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
22	21	adantl 271	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( A .Q B ) = 1Q <-> ( B e. Q. /\ ( A .Q B ) = 1Q ) ) )
23	20, 22	bitr4d 189	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   = wceq 1289   E.wex 1426   e. wcel 1438   E!weu 1948   E*wmo 1949   {copab 3898   ` cfv 5015  (class class class)co 5652   Q.cnq 6839   1Qc1q 6840   .Q cmq 6842   *Qcrq 6843

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-mi 6865  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-mqqs 6909  df-1nqqs 6910  df-rq 6911

This theorem is referenced by:  recclnq  6951  recidnq  6952  recrecnq  6953  recexprlem1ssl  7192  recexprlem1ssu  7193