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Theorem recmulnqg 7199
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.
StepHypRef Expression
1 oveq1 5781 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
21eqeq1d 2148 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
32anbi2d 459 . . 3  |-  ( x  =  A  ->  (
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( y  e.  Q.  /\  ( A  .Q  y
)  =  1Q ) ) )
4 eleq1 2202 . . . 4  |-  ( y  =  B  ->  (
y  e.  Q.  <->  B  e.  Q. ) )
5 oveq2 5782 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
65eqeq1d 2148 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
74, 6anbi12d 464 . . 3  |-  ( y  =  B  ->  (
( y  e.  Q.  /\  ( A  .Q  y
)  =  1Q )  <-> 
( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
8 recexnq 7198 . . . 4  |-  ( x  e.  Q.  ->  E. y
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9 1nq 7174 . . . . 5  |-  1Q  e.  Q.
10 mulcomnqg 7191 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  .Q  w
)  =  ( w  .Q  z ) )
11 mulassnqg 7192 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )  ->  (
( z  .Q  w
)  .Q  v )  =  ( z  .Q  ( w  .Q  v
) ) )
12 mulidnq 7197 . . . . 5  |-  ( z  e.  Q.  ->  (
z  .Q  1Q )  =  z )
139, 10, 11, 12caovimo 5964 . . . 4  |-  ( x  e.  Q.  ->  E* y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
14 eu5 2046 . . . 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 ) ) )
158, 13, 14sylanbrc 413 . . 3  |-  ( x  e.  Q.  ->  E! y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
16 df-rq 7160 . . . 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 ) ) )
1817opabbii 3995 . . . 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 ) ) }
1916, 18eqtri 2160 . . 3  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) }
203, 7, 15, 19fvopab3g 5494 . 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 ) ) )
2221adantl 275 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( A  .Q  B )  =  1Q  <->  ( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
2320, 22bitr4d 190 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( *Q `  A )  =  B  <-> 
( A  .Q  B
)  =  1Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1999   E*wmo 2000   {copab 3988   ` cfv 5123  (class class class)co 5774   Q.cnq 7088   1Qc1q 7089    .Q cmq 7091   *Qcrq 7092
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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-mi 7114  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-mqqs 7158  df-1nqqs 7159  df-rq 7160
This theorem is referenced by:  recclnq  7200  recidnq  7201  recrecnq  7202  recexprlem1ssl  7441  recexprlem1ssu  7442
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