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Theorem recidnq 8584
Description: A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
recidnq  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )

Proof of Theorem recidnq
StepHypRef Expression
1 eqid 2283 . 2  |-  ( *Q
`  A )  =  ( *Q `  A
)
2 recmulnq 8583 . 2  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  ( *Q
`  A )  <->  ( A  .Q  ( *Q `  A
) )  =  1Q ) )
31, 2mpbii 202 1  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5220  (class class class)co 5819   Q.cnq 8469   1Qc1q 8470    .Q cmq 8473   *Qcrq 8474
This theorem is referenced by:  recclnq  8585  recrecnq  8586  dmrecnq  8587  halfnq  8595  ltrnq  8598  addclprlem1  8635  addclprlem2  8636  mulclprlem  8638  1idpr  8648  prlem934  8652  prlem936  8666  reclem3pr  8668  reclem4pr  8669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4186  ax-pr 4212  ax-un 4510
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-we 4352  df-ord 4393  df-on 4394  df-lim 4395  df-suc 4396  df-om 4655  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  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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