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Theorem recclnq 8843
Description: Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
recclnq  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
Q. )

Proof of Theorem recclnq
StepHypRef Expression
1 recidnq 8842 . . . 4  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
2 1nq 8805 . . . 4  |-  1Q  e.  Q.
31, 2syl6eqel 2524 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  e. 
Q. )
4 mulnqf 8826 . . . . 5  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5596 . . . 4  |-  dom  .Q  =  ( Q.  X.  Q. )
6 0nnq 8801 . . . 4  |-  -.  (/)  e.  Q.
75, 6ndmovrcl 6233 . . 3  |-  ( ( A  .Q  ( *Q
`  A ) )  e.  Q.  ->  ( A  e.  Q.  /\  ( *Q `  A )  e. 
Q. ) )
83, 7syl 16 . 2  |-  ( A  e.  Q.  ->  ( A  e.  Q.  /\  ( *Q `  A )  e. 
Q. ) )
98simprd 450 1  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    X. cxp 4876   ` cfv 5454  (class class class)co 6081   Q.cnq 8727   1Qc1q 8728    .Q cmq 8731   *Qcrq 8732
This theorem is referenced by:  recrecnq  8844  dmrecnq  8845  halfnq  8853  ltrnq  8856  mulclprlem  8896  prlem934  8910  prlem936  8924  reclem2pr  8925  reclem3pr  8926  reclem4pr  8927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-mi 8751  df-lti 8752  df-mpq 8786  df-enq 8788  df-nq 8789  df-erq 8790  df-mq 8792  df-1nq 8793  df-rq 8794
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