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Theorem recrecnq 9242
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 5794 . . . 4  |-  ( x  =  A  ->  ( *Q `  x )  =  ( *Q `  A
) )
21fveq2d 5798 . . 3  |-  ( x  =  A  ->  ( *Q `  ( *Q `  x ) )  =  ( *Q `  ( *Q `  A ) ) )
3 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2474 . 2  |-  ( x  =  A  ->  (
( *Q `  ( *Q `  x ) )  =  x  <->  ( *Q `  ( *Q `  A
) )  =  A ) )
5 mulcomnq 9228 . . . 4  |-  ( ( *Q `  x )  .Q  x )  =  ( x  .Q  ( *Q `  x ) )
6 recidnq 9240 . . . 4  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
75, 6syl5eq 2505 . . 3  |-  ( x  e.  Q.  ->  (
( *Q `  x
)  .Q  x )  =  1Q )
8 recclnq 9241 . . . 4  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
9 recmulnq 9239 . . . 4  |-  ( ( *Q `  x )  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  =  x  <->  ( ( *Q `  x )  .Q  x )  =  1Q ) )
108, 9syl 16 . . 3  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  =  x  <->  ( ( *Q `  x )  .Q  x )  =  1Q ) )
117, 10mpbird 232 . 2  |-  ( x  e.  Q.  ->  ( *Q `  ( *Q `  x ) )  =  x )
124, 11vtoclga 3136 1  |-  ( A  e.  Q.  ->  ( *Q `  ( *Q `  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   ` cfv 5521  (class class class)co 6195   Q.cnq 9125   1Qc1q 9126    .Q cmq 9129   *Qcrq 9130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-omul 7030  df-er 7206  df-ni 9147  df-mi 9149  df-lti 9150  df-mpq 9184  df-enq 9186  df-nq 9187  df-erq 9188  df-mq 9190  df-1nq 9191  df-rq 9192
This theorem is referenced by:  reclem2pr  9323
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