Theorem recmulnqg 7706

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 6057	. . . . 5 |- ( x = A -> ( x .Q y ) = ( A .Q y ) )
2	1	eqeq1d 2241	. . . 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 2295	. . . 4 |- ( y = B -> ( y e. Q. <-> B e. Q. ) )
5		oveq2 6058	. . . . 5 |- ( y = B -> ( A .Q y ) = ( A .Q B ) )
6	5	eqeq1d 2241	. . . 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 7705	. . . 4 |- ( x e. Q. -> E. y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
9		1nq 7681	. . . . 5 |- 1Q e. Q.
10		mulcomnqg 7698	. . . . 5 |- ( ( z e. Q. /\ w e. Q. ) -> ( z .Q w ) = ( w .Q z ) )
11		mulassnqg 7699	. . . . 5 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( ( z .Q w ) .Q v ) = ( z .Q ( w .Q v ) ) )
12		mulidnq 7704	. . . . 5 |- ( z e. Q. -> ( z .Q 1Q ) = z )
13	9, 10, 11, 12	caovimo 6248	. . . 4 |- ( x e. Q. -> E* y ( y e. Q. /\ ( x .Q y ) = 1Q ) )
14		eu5 2128	. . . 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 7667	. . . 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 4177	. . . 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 2253	. . 3 |- *Q = { <. x , y >. | ( x e. Q. /\ ( y e. Q. /\ ( x .Q y ) = 1Q ) ) }
20	3, 7, 15, 19	fvopab3g 5750	. 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 2080   E*wmo 2081   e. wcel 2203   {copab 4170   ` cfv 5352  (class class class)co 6050   Q.cnq 7595   1Qc1q 7596   .Q cmq 7598   *Qcrq 7599

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-mi 7621  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-mqqs 7665  df-1nqqs 7666  df-rq 7667

This theorem is referenced by:  recclnq  7707  recidnq  7708  recrecnq  7709  recexprlem1ssl  7948  recexprlem1ssu  7949