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Theorem recidnq 8843
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 2437 . 2  |-  ( *Q
`  A )  =  ( *Q `  A
)
2 recmulnq 8842 . 2  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  ( *Q
`  A )  <->  ( A  .Q  ( *Q `  A
) )  =  1Q ) )
31, 2mpbii 204 1  |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   Q.cnq 8728   1Qc1q 8729    .Q cmq 8732   *Qcrq 8733
This theorem is referenced by:  recclnq  8844  recrecnq  8845  dmrecnq  8846  halfnq  8854  ltrnq  8857  addclprlem1  8894  addclprlem2  8895  mulclprlem  8897  1idpr  8907  prlem934  8911  prlem936  8925  reclem3pr  8927  reclem4pr  8928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-omul 6730  df-er 6906  df-ni 8750  df-mi 8752  df-lti 8753  df-mpq 8787  df-enq 8789  df-nq 8790  df-erq 8791  df-mq 8793  df-1nq 8794  df-rq 8795
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