ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rerecclap Unicode version

Theorem rerecclap 8718
Description: Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
rerecclap  |-  ( ( A  e.  RR  /\  A #  0 )  ->  (
1  /  A )  e.  RR )

Proof of Theorem rerecclap
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0re 7988 . . . . . 6  |-  0  e.  RR
2 apreap 8575 . . . . . 6  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  A #  0
) )
31, 2mpan2 425 . . . . 5  |-  ( A  e.  RR  ->  ( A #  0  <->  A #  0 ) )
43pm5.32i 454 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( A  e.  RR  /\  A #  0 ) )
5 recexre 8566 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
64, 5sylbi 121 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
7 eqcom 2191 . . . . 5  |-  ( x  =  ( 1  /  A )  <->  ( 1  /  A )  =  x )
8 1cnd 8004 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  1  e.  CC )
9 simpr 110 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  x  e.  RR )
109recnd 8017 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  x  e.  CC )
11 simpll 527 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A  e.  RR )
1211recnd 8017 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A  e.  CC )
13 simplr 528 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A #  0 )
14 divmulap 8663 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC  /\  ( A  e.  CC  /\  A #  0 ) )  -> 
( ( 1  /  A )  =  x  <-> 
( A  x.  x
)  =  1 ) )
158, 10, 12, 13, 14syl112anc 1253 . . . . 5  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  ( ( 1  /  A )  =  x  <-> 
( A  x.  x
)  =  1 ) )
167, 15bitrid 192 . . . 4  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  ( x  =  ( 1  /  A )  <-> 
( A  x.  x
)  =  1 ) )
1716rexbidva 2487 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( E. x  e.  RR  x  =  ( 1  /  A )  <->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
186, 17mpbird 167 . 2  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  x  =  ( 1  /  A ) )
19 risset 2518 . 2  |-  ( ( 1  /  A )  e.  RR  <->  E. x  e.  RR  x  =  ( 1  /  A ) )
2018, 19sylibr 134 1  |-  ( ( A  e.  RR  /\  A #  0 )  ->  (
1  /  A )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   E.wrex 2469   class class class wbr 4018  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842   1c1 7843    x. cmul 7847   # creap 8562   # cap 8569    / cdiv 8660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661
This theorem is referenced by:  redivclap  8719  rerecclapzi  8764  rerecclapd  8822  rerecapb  8831  ltdiv2  8875  recnz  9377  reexpclzap  10574  redivap  10918  imdivap  10925  caucvgrelemrec  11023  trirec0  15271
  Copyright terms: Public domain W3C validator