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Theorem rerecclap 8640
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 7913 . . . . . 6  |-  0  e.  RR
2 apreap 8499 . . . . . 6  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  A #  0
) )
31, 2mpan2 423 . . . . 5  |-  ( A  e.  RR  ->  ( A #  0  <->  A #  0 ) )
43pm5.32i 451 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( A  e.  RR  /\  A #  0 ) )
5 recexre 8490 . . . 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 2172 . . . . 5  |-  ( x  =  ( 1  /  A )  <->  ( 1  /  A )  =  x )
8 1cnd 7929 . . . . . 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 7941 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  x  e.  CC )
11 simpll 524 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A  e.  RR )
1211recnd 7941 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A  e.  CC )
13 simplr 525 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  x  e.  RR )  ->  A #  0 )
14 divmulap 8585 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC  /\  ( A  e.  CC  /\  A #  0 ) )  -> 
( ( 1  /  A )  =  x  <-> 
( A  x.  x
)  =  1 ) )
158, 10, 12, 13, 14syl112anc 1237 . . . . 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 2467 . . 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 2498 . 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 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3987  (class class class)co 5851   CCcc 7765   RRcr 7766   0cc0 7767   1c1 7768    x. cmul 7772   # creap 8486   # cap 8493    / cdiv 8582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583
This theorem is referenced by:  redivclap  8641  rerecclapzi  8686  rerecclapd  8744  ltdiv2  8796  recnz  9298  reexpclzap  10489  redivap  10831  imdivap  10838  caucvgrelemrec  10936  trirec0  14041
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