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Theorem rerecclap 8294
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 7585 . . . . . 6  |-  0  e.  RR
2 apreap 8161 . . . . . 6  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  A #  0
) )
31, 2mpan2 417 . . . . 5  |-  ( A  e.  RR  ->  ( A #  0  <->  A #  0 ) )
43pm5.32i 443 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( A  e.  RR  /\  A #  0 ) )
5 recexre 8152 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
64, 5sylbi 120 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
7 eqcom 2097 . . . . 5  |-  ( x  =  ( 1  /  A )  <->  ( 1  /  A )  =  x )
8 1cnd 7601 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  1  e.  CC )
9 simpr 109 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  x  e.  RR )
109recnd 7613 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  x  e.  CC )
11 simpll 497 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A  e.  RR )
1211recnd 7613 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A  e.  CC )
13 simplr 498 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A #  0 )
14 divmulap 8239 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC  /\  ( A  e.  CC  /\  A #  0 ) )  -> 
( ( 1  /  A )  =  x  <-> 
( A  x.  x
)  =  1 ) )
158, 10, 12, 13, 14syl112anc 1185 . . . . 5  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  ( ( 1  /  A )  =  x  <-> 
( A  x.  x
)  =  1 ) )
167, 15syl5bb 191 . . . 4  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  ( x  =  ( 1  /  A )  <-> 
( A  x.  x
)  =  1 ) )
1716rexbidva 2388 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( E. x  e.  RR  x  =  ( 1  /  A )  <->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
186, 17mpbird 166 . 2  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  x  =  ( 1  /  A ) )
19 risset 2417 . 2  |-  ( ( 1  /  A )  e.  RR  <->  E. x  e.  RR  x  =  ( 1  /  A ) )
2018, 19sylibr 133 1  |-  ( ( A  e.  RR  /\  A #  0 )  ->  (
1  /  A )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448    x. cmul 7452   # creap 8148   # cap 8155    / cdiv 8236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237
This theorem is referenced by:  redivclap  8295  rerecclapzi  8340  rerecclapd  8397  ltdiv2  8445  recnz  8938  reexpclzap  10090  redivap  10423  imdivap  10430  caucvgrelemrec  10527
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