ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recmulnqg Unicode version

Theorem recmulnqg 6547
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 5547 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
21eqeq1d 2064 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
32anbi2d 445 . . 3  |-  ( x  =  A  ->  (
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( y  e.  Q.  /\  ( A  .Q  y
)  =  1Q ) ) )
4 eleq1 2116 . . . 4  |-  ( y  =  B  ->  (
y  e.  Q.  <->  B  e.  Q. ) )
5 oveq2 5548 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
65eqeq1d 2064 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
74, 6anbi12d 450 . . 3  |-  ( y  =  B  ->  (
( y  e.  Q.  /\  ( A  .Q  y
)  =  1Q )  <-> 
( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
8 recexnq 6546 . . . 4  |-  ( x  e.  Q.  ->  E. y
( y  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9 1nq 6522 . . . . 5  |-  1Q  e.  Q.
10 mulcomnqg 6539 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  .Q  w
)  =  ( w  .Q  z ) )
11 mulassnqg 6540 . . . . 5  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )  ->  (
( z  .Q  w
)  .Q  v )  =  ( z  .Q  ( w  .Q  v
) ) )
12 mulidnq 6545 . . . . 5  |-  ( z  e.  Q.  ->  (
z  .Q  1Q )  =  z )
139, 10, 11, 12caovimo 5722 . . . 4  |-  ( x  e.  Q.  ->  E* y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
14 eu5 1963 . . . 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 402 . . 3  |-  ( x  e.  Q.  ->  E! y ( y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
16 df-rq 6508 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  y  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) }
17 3anass 900 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  (
x  .Q  y )  =  1Q )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) )
1817opabbii 3852 . . . 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 2076 . . 3  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( y  e.  Q.  /\  (
x  .Q  y )  =  1Q ) ) }
203, 7, 15, 19fvopab3g 5273 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( *Q `  A )  =  B  <-> 
( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
21 ibar 289 . . 3  |-  ( B  e.  Q.  ->  (
( A  .Q  B
)  =  1Q  <->  ( B  e.  Q.  /\  ( A  .Q  B )  =  1Q ) ) )
2221adantl 266 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( A  .Q  B )  =  1Q  <->  ( B  e.  Q.  /\  ( A  .Q  B
)  =  1Q ) ) )
2320, 22bitr4d 184 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 101    <-> wb 102    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   E!weu 1916   E*wmo 1917   {copab 3845   ` cfv 4930  (class class class)co 5540   Q.cnq 6436   1Qc1q 6437    .Q cmq 6439   *Qcrq 6440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-mqqs 6506  df-1nqqs 6507  df-rq 6508
This theorem is referenced by:  recclnq  6548  recidnq  6549  recrecnq  6550  recexprlem1ssl  6789  recexprlem1ssu  6790
  Copyright terms: Public domain W3C validator