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Theorem recrecnq 8591
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  |-  ( A  e.  Q.  ->  ( *Q `  ( *Q `  A ) )  =  A )

Proof of Theorem recrecnq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( x  =  A  ->  ( *Q `  x )  =  ( *Q `  A
) )
21fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( *Q `  ( *Q `  x ) )  =  ( *Q `  ( *Q `  A ) ) )
3 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2297 . 2  |-  ( x  =  A  ->  (
( *Q `  ( *Q `  x ) )  =  x  <->  ( *Q `  ( *Q `  A
) )  =  A ) )
5 mulcomnq 8577 . . . 4  |-  ( ( *Q `  x )  .Q  x )  =  ( x  .Q  ( *Q `  x ) )
6 recidnq 8589 . . . 4  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
75, 6syl5eq 2327 . . 3  |-  ( x  e.  Q.  ->  (
( *Q `  x
)  .Q  x )  =  1Q )
8 recclnq 8590 . . . 4  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
9 recmulnq 8588 . . . 4  |-  ( ( *Q `  x )  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  =  x  <->  ( ( *Q `  x )  .Q  x )  =  1Q ) )
108, 9syl 15 . . 3  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  =  x  <->  ( ( *Q `  x )  .Q  x )  =  1Q ) )
117, 10mpbird 223 . 2  |-  ( x  e.  Q.  ->  ( *Q `  ( *Q `  x ) )  =  x )
124, 11vtoclga 2849 1  |-  ( A  e.  Q.  ->  ( *Q `  ( *Q `  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Q.cnq 8474   1Qc1q 8475    .Q cmq 8478   *Qcrq 8479
This theorem is referenced by:  reclem2pr  8672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-mpq 8533  df-enq 8535  df-nq 8536  df-erq 8537  df-mq 8539  df-1nq 8540  df-rq 8541
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