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Theorem recidnq 8591
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 2285 . 2  |-  ( *Q
`  A )  =  ( *Q `  A
)
2 recmulnq 8590 . 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 1625    e. wcel 1686   ` cfv 5257  (class class class)co 5860   Q.cnq 8476   1Qc1q 8477    .Q cmq 8480   *Qcrq 8481
This theorem is referenced by:  recclnq  8592  recrecnq  8593  dmrecnq  8594  halfnq  8602  ltrnq  8605  addclprlem1  8642  addclprlem2  8643  mulclprlem  8645  1idpr  8655  prlem934  8659  prlem936  8673  reclem3pr  8675  reclem4pr  8676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-ni 8498  df-mi 8500  df-lti 8501  df-mpq 8535  df-enq 8537  df-nq 8538  df-erq 8539  df-mq 8541  df-1nq 8542  df-rq 8543
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