Theorem recmulnqg 7475

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 5932	. . . . 5 |- ( x = A -> ( x .Q y ) = ( A .Q y ) )
2	1	eqeq1d 2205	. . . 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 2259	. . . 4 |- ( y = B -> ( y e. Q. <-> B e. Q. ) )
5		oveq2 5933	. . . . 5 |- ( y = B -> ( A .Q y ) = ( A .Q B ) )
6	5	eqeq1d 2205	. . . 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 7474	. . . 4 |- ( x e. Q. -> E. y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
9		1nq 7450	. . . . 5 |- 1Q e. Q.
10		mulcomnqg 7467	. . . . 5 |- ( ( z e. Q. /\ w e. Q. ) -> ( z .Q w ) = ( w .Q z ) )
11		mulassnqg 7468	. . . . 5 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( ( z .Q w ) .Q v ) = ( z .Q ( w .Q v ) ) )
12		mulidnq 7473	. . . . 5 |- ( z e. Q. -> ( z .Q 1Q ) = z )
13	9, 10, 11, 12	caovimo 6121	. . . 4 |- ( x e. Q. -> E* y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
14		eu5 2092	. . . 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 7436	. . . 4 |- *Q = { <. x , y >. | ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) }
17		3anass 984	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) <-> ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) )
18	17	opabbii 4101	. . . 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 2217	. . 3 |- *Q = { <. x , y >. | ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) }
20	3, 7, 15, 19	fvopab3g 5637	. 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 980   = wceq 1364   E.wex 1506   E!weu 2045   E*wmo 2046   e. wcel 2167   {copab 4094   ` cfv 5259  (class class class)co 5925   Q.cnq 7364   1Qc1q 7365   .Q cmq 7367   *Qcrq 7368

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-mi 7390  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-mqqs 7434  df-1nqqs 7435  df-rq 7436

This theorem is referenced by:  recclnq  7476  recidnq  7477  recrecnq  7478  recexprlem1ssl  7717  recexprlem1ssu  7718