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Theorem recrec 9695
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 9677 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
2 ax-1cn 9032 . . . 4  |-  1  e.  CC
32a1i 11 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
1  e.  CC )
4 simpl 444 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
5 reccl 9669 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
6 recne0 9675 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
7 divmul 9665 . . 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 1188 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  / 
( 1  /  A
) )  =  A  <-> 
( ( 1  /  A )  x.  A
)  =  1 ) )
91, 8mpbird 224 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2593  (class class class)co 6067   CCcc 8972   0cc0 8974   1c1 8975    x. cmul 8979    / cdiv 9661
This theorem is referenced by:  recreci  9730  recrecd  9771  ltrec1  9881  lerec2  9882  resqrex  12039  logrec  20644  rlimcnp  20787  rlimcnp2  20788  recsec  28255  reccsc  28256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-po 4490  df-so 4491  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-riota 6535  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662
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