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Theorem recidnq 9731
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 (𝐴Q → (𝐴 ·Q (*Q𝐴)) = 1Q)

Proof of Theorem recidnq
StepHypRef Expression
1 eqid 2621 . 2 (*Q𝐴) = (*Q𝐴)
2 recmulnq 9730 . 2 (𝐴Q → ((*Q𝐴) = (*Q𝐴) ↔ (𝐴 ·Q (*Q𝐴)) = 1Q))
31, 2mpbii 223 1 (𝐴Q → (𝐴 ·Q (*Q𝐴)) = 1Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  Qcnq 9618  1Qc1q 9619   ·Q cmq 9622  *Qcrq 9623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-omul 7510  df-er 7687  df-ni 9638  df-mi 9640  df-lti 9641  df-mpq 9675  df-enq 9677  df-nq 9678  df-erq 9679  df-mq 9681  df-1nq 9682  df-rq 9683
This theorem is referenced by:  recclnq  9732  recrecnq  9733  dmrecnq  9734  halfnq  9742  ltrnq  9745  addclprlem1  9782  addclprlem2  9783  mulclprlem  9785  1idpr  9795  prlem934  9799  prlem936  9813  reclem3pr  9815  reclem4pr  9816
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