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Theorem recidnq 8557
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 2258 . 2  |-  ( *Q
`  A )  =  ( *Q `  A
)
2 recmulnq 8556 . 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 6    = wceq 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   Q.cnq 8442   1Qc1q 8443    .Q cmq 8446   *Qcrq 8447
This theorem is referenced by:  recclnq  8558  recrecnq  8559  dmrecnq  8560  halfnq  8568  ltrnq  8571  addclprlem1  8608  addclprlem2  8609  mulclprlem  8611  1idpr  8621  prlem934  8625  prlem936  8639  reclem3pr  8641  reclem4pr  8642
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-mi 8466  df-lti 8467  df-mpq 8501  df-enq 8503  df-nq 8504  df-erq 8505  df-mq 8507  df-1nq 8508  df-rq 8509
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