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Theorem rereccl 10161
Description: Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
rereccl  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )

Proof of Theorem rereccl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-rrecex 9466 . . 3  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
2 eqcom 2463 . . . . 5  |-  ( x  =  ( 1  /  A )  <->  ( 1  /  A )  =  x )
3 ax-1cn 9452 . . . . . . 7  |-  1  e.  CC
43a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  1  e.  CC )
5 simpr 461 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  x  e.  RR )
65recnd 9524 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  x  e.  CC )
7 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  A  e.  RR )
87recnd 9524 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  A  e.  CC )
9 simplr 754 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  A  =/=  0
)
10 divmul 10109 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( 1  /  A )  =  x  <-> 
( A  x.  x
)  =  1 ) )
114, 6, 8, 9, 10syl112anc 1223 . . . . 5  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  ( ( 1  /  A )  =  x  <->  ( A  x.  x )  =  1 ) )
122, 11syl5bb 257 . . . 4  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  x  e.  RR )  ->  ( x  =  ( 1  /  A
)  <->  ( A  x.  x )  =  1 ) )
1312rexbidva 2865 . . 3  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( E. x  e.  RR  x  =  ( 1  /  A )  <->  E. x  e.  RR  ( A  x.  x
)  =  1 ) )
141, 13mpbird 232 . 2  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  E. x  e.  RR  x  =  ( 1  /  A ) )
15 risset 2885 . 2  |-  ( ( 1  /  A )  e.  RR  <->  E. x  e.  RR  x  =  ( 1  /  A ) )
1614, 15sylibr 212 1  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800  (class class class)co 6201   CCcc 9392   RRcr 9393   0cc0 9394   1c1 9395    x. cmul 9399    / cdiv 10105
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106
This theorem is referenced by:  redivcl  10162  rerecclzi  10207  rereccld  10270  ltdiv2  10329  ltrec1  10331  lerec2  10332  lediv2  10334  lediv12a  10337  recreclt  10343  recnz  10829  reexpclz  12003  rediv  12739  imdiv  12746  resqrex  12859  resubdrg  18164  axcontlem2  23364  leopmul  25691  nmopleid  25696  cdj1i  25990  lediv2aALT  27467
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