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Theorem recclnq 8585
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 8584 . . . 4  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
2 1nq 8547 . . . 4  |-  1Q  e.  Q.
31, 2syl6eqel 2372 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  e. 
Q. )
4 mulnqf 8568 . . . . 5  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5359 . . . 4  |-  dom  .Q  =  ( Q.  X.  Q. )
6 0nnq 8543 . . . 4  |-  -.  (/)  e.  Q.
75, 6ndmovrcl 5967 . . 3  |-  ( ( A  .Q  ( *Q
`  A ) )  e.  Q.  ->  ( A  e.  Q.  /\  ( *Q `  A )  e. 
Q. ) )
83, 7syl 17 . 2  |-  ( A  e.  Q.  ->  ( A  e.  Q.  /\  ( *Q `  A )  e. 
Q. ) )
98simprd 451 1  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1685    X. cxp 4686   ` cfv 5221  (class class class)co 5819   Q.cnq 8469   1Qc1q 8470    .Q cmq 8473   *Qcrq 8474
This theorem is referenced by:  recrecnq  8586  dmrecnq  8587  halfnq  8595  ltrnq  8598  mulclprlem  8638  prlem934  8652  prlem936  8666  reclem2pr  8667  reclem3pr  8668  reclem4pr  8669
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-mi 8493  df-lti 8494  df-mpq 8528  df-enq 8530  df-nq 8531  df-erq 8532  df-mq 8534  df-1nq 8535  df-rq 8536
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