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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.
StepHypRef Expression
1 oveq1 5659 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
21eqeq1d 2096 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
32anbi2d 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
) )
65eqeq1d 2096 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
74, 6anbi12d 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 )
139, 10, 11, 12caovimo 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 ) ) )
158, 13, 14sylanbrc 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 ) ) )
1817opabbii 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 ) ) }
1916, 18eqtri 2108 . . 3  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) }
203, 7, 15, 19fvopab3g 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 ) ) )
2221adantl 271 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( A  .Q  B )  =  1Q  <->  ( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
2320, 22bitr4d 189 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 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
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