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Theorem recmulnqg 7167
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 5749 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
21eqeq1d 2126 . . . 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 2180 . . . 4  |-  ( y  =  B  ->  (
y  e.  Q.  <->  B  e.  Q. ) )
5 oveq2 5750 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
65eqeq1d 2126 . . . 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 7166 . . . 4  |-  ( x  e.  Q.  ->  E. y
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9 1nq 7142 . . . . 5  |-  1Q  e.  Q.
10 mulcomnqg 7159 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  .Q  w
)  =  ( w  .Q  z ) )
11 mulassnqg 7160 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )  ->  (
( z  .Q  w
)  .Q  v )  =  ( z  .Q  ( w  .Q  v
) ) )
12 mulidnq 7165 . . . . 5  |-  ( z  e.  Q.  ->  (
z  .Q  1Q )  =  z )
139, 10, 11, 12caovimo 5932 . . . 4  |-  ( x  e.  Q.  ->  E* y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
14 eu5 2024 . . . 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 7128 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) }
17 3anass 951 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  (
x  .Q  y )  =  1Q )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) )
1817opabbii 3965 . . . 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 2138 . . 3  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) }
203, 7, 15, 19fvopab3g 5462 . 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 947    = wceq 1316   E.wex 1453    e. wcel 1465   E!weu 1977   E*wmo 1978   {copab 3958   ` cfv 5093  (class class class)co 5742   Q.cnq 7056   1Qc1q 7057    .Q cmq 7059   *Qcrq 7060
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-mi 7082  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-mqqs 7126  df-1nqqs 7127  df-rq 7128
This theorem is referenced by:  recclnq  7168  recidnq  7169  recrecnq  7170  recexprlem1ssl  7409  recexprlem1ssu  7410
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