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Theorem recrec 9452
Description: A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 26-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
recrec  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )

Proof of Theorem recrec
StepHypRef Expression
1 recid2 9434 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
2 ax-1cn 8790 . . . 4  |-  1  e.  CC
32a1i 12 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
1  e.  CC )
4 simpl 445 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
5 reccl 9426 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
6 recne0 9432 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
7 divmul 9422 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  (
( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 ) )  ->  ( (
1  /  ( 1  /  A ) )  =  A  <->  ( (
1  /  A )  x.  A )  =  1 ) )
83, 4, 5, 6, 7syl112anc 1191 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  / 
( 1  /  A
) )  =  A  <-> 
( ( 1  /  A )  x.  A
)  =  1 ) )
91, 8mpbird 225 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1628    e. wcel 1688    =/= wne 2447  (class class class)co 5819   CCcc 8730   0cc0 8732   1c1 8733    x. cmul 8737    / cdiv 9418
This theorem is referenced by:  recreci  9487  recrecd  9528  ltrec1  9638  lerec2  9639  resqrex  11730  logrec  20111  rlimcnp  20254  rlimcnp2  20255  recsec  27486  reccsc  27487
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419
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