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Theorem recmulnqg 7421
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 5904 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
21eqeq1d 2198 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
32anbi2d 464 . . 3  |-  ( x  =  A  ->  (
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( y  e.  Q.  /\  ( A  .Q  y
)  =  1Q ) ) )
4 eleq1 2252 . . . 4  |-  ( y  =  B  ->  (
y  e.  Q.  <->  B  e.  Q. ) )
5 oveq2 5905 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
65eqeq1d 2198 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
74, 6anbi12d 473 . . 3  |-  ( y  =  B  ->  (
( y  e.  Q.  /\  ( A  .Q  y
)  =  1Q )  <-> 
( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
8 recexnq 7420 . . . 4  |-  ( x  e.  Q.  ->  E. y
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9 1nq 7396 . . . . 5  |-  1Q  e.  Q.
10 mulcomnqg 7413 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  .Q  w
)  =  ( w  .Q  z ) )
11 mulassnqg 7414 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )  ->  (
( z  .Q  w
)  .Q  v )  =  ( z  .Q  ( w  .Q  v
) ) )
12 mulidnq 7419 . . . . 5  |-  ( z  e.  Q.  ->  (
z  .Q  1Q )  =  z )
139, 10, 11, 12caovimo 6091 . . . 4  |-  ( x  e.  Q.  ->  E* y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
14 eu5 2085 . . . 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 417 . . 3  |-  ( x  e.  Q.  ->  E! y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
16 df-rq 7382 . . . 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 ) ) )
1817opabbii 4085 . . . 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 2210 . . 3  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) }
203, 7, 15, 19fvopab3g 5610 . 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 ) ) )
2221adantl 277 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( A  .Q  B )  =  1Q  <->  ( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
2320, 22bitr4d 191 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 104    <-> wb 105    /\ w3a 980    = wceq 1364   E.wex 1503   E!weu 2038   E*wmo 2039    e. wcel 2160   {copab 4078   ` cfv 5235  (class class class)co 5897   Q.cnq 7310   1Qc1q 7311    .Q cmq 7313   *Qcrq 7314
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-mi 7336  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-mqqs 7380  df-1nqqs 7381  df-rq 7382
This theorem is referenced by:  recclnq  7422  recidnq  7423  recrecnq  7424  recexprlem1ssl  7663  recexprlem1ssu  7664
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