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Theorem recrecnq 10380
 Description: Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
recrecnq (𝐴Q → (*Q‘(*Q𝐴)) = 𝐴)

Proof of Theorem recrecnq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 2fveq3 6650 . . 3 (𝑥 = 𝐴 → (*Q‘(*Q𝑥)) = (*Q‘(*Q𝐴)))
2 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
31, 2eqeq12d 2814 . 2 (𝑥 = 𝐴 → ((*Q‘(*Q𝑥)) = 𝑥 ↔ (*Q‘(*Q𝐴)) = 𝐴))
4 mulcomnq 10366 . . . 4 ((*Q𝑥) ·Q 𝑥) = (𝑥 ·Q (*Q𝑥))
5 recidnq 10378 . . . 4 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
64, 5syl5eq 2845 . . 3 (𝑥Q → ((*Q𝑥) ·Q 𝑥) = 1Q)
7 recclnq 10379 . . . 4 (𝑥Q → (*Q𝑥) ∈ Q)
8 recmulnq 10377 . . . 4 ((*Q𝑥) ∈ Q → ((*Q‘(*Q𝑥)) = 𝑥 ↔ ((*Q𝑥) ·Q 𝑥) = 1Q))
97, 8syl 17 . . 3 (𝑥Q → ((*Q‘(*Q𝑥)) = 𝑥 ↔ ((*Q𝑥) ·Q 𝑥) = 1Q))
106, 9mpbird 260 . 2 (𝑥Q → (*Q‘(*Q𝑥)) = 𝑥)
113, 10vtoclga 3522 1 (𝐴Q → (*Q‘(*Q𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  Qcnq 10265  1Qc1q 10266   ·Q cmq 10269  *Qcrq 10270 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-omul 8092  df-er 8274  df-ni 10285  df-mi 10287  df-lti 10288  df-mpq 10322  df-enq 10324  df-nq 10325  df-erq 10326  df-mq 10328  df-1nq 10329  df-rq 10330 This theorem is referenced by:  reclem2pr  10461
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