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Theorem recexre 8303
Description: Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
recexre  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
Distinct variable group:    x, A

Proof of Theorem recexre
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 0re 7730 . . . 4  |-  0  e.  RR
2 reapval 8301 . . . 4  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  ( A  <  0  \/  0  < 
A ) ) )
31, 2mpan2 419 . . 3  |-  ( A  e.  RR  ->  ( A #  0 
<->  ( A  <  0  \/  0  <  A ) ) )
4 lt0neg1 8194 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
5 renegcl 7987 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 ltxrlt 7794 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  -u A  e.  RR )  ->  ( 0  <  -u A  <->  0  <RR  -u A
) )
71, 5, 6sylancr 408 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
0  <  -u A  <->  0  <RR  -u A ) )
84, 7bitrd 187 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <RR  -u A ) )
98pm5.32i 447 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  0 )  <->  ( A  e.  RR  /\  0  <RR  -u A ) )
10 ax-precex 7694 . . . . . . . . . 10  |-  ( (
-u A  e.  RR  /\  0  <RR  -u A )  ->  E. y  e.  RR  ( 0  <RR  y  /\  ( -u A  x.  y
)  =  1 ) )
11 simpr 109 . . . . . . . . . . 11  |-  ( ( 0  <RR  y  /\  ( -u A  x.  y )  =  1 )  -> 
( -u A  x.  y
)  =  1 )
1211reximi 2504 . . . . . . . . . 10  |-  ( E. y  e.  RR  (
0  <RR  y  /\  ( -u A  x.  y )  =  1 )  ->  E. y  e.  RR  ( -u A  x.  y
)  =  1 )
1310, 12syl 14 . . . . . . . . 9  |-  ( (
-u A  e.  RR  /\  0  <RR  -u A )  ->  E. y  e.  RR  ( -u A  x.  y
)  =  1 )
145, 13sylan 279 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <RR  -u A )  ->  E. y  e.  RR  ( -u A  x.  y
)  =  1 )
159, 14sylbi 120 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  E. y  e.  RR  ( -u A  x.  y
)  =  1 )
16 recn 7717 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  y  e.  CC )
1716negnegd 8028 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  -u -u y  =  y )
1817oveq2d 5756 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  ( -u A  x.  -u -u y
)  =  ( -u A  x.  y )
)
1918eqeq1d 2124 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
( -u A  x.  -u -u y
)  =  1  <->  ( -u A  x.  y )  =  1 ) )
2019pm5.32i 447 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  ( -u A  x.  -u -u y
)  =  1 )  <-> 
( y  e.  RR  /\  ( -u A  x.  y )  =  1 ) )
21 renegcl 7987 . . . . . . . . . 10  |-  ( y  e.  RR  ->  -u y  e.  RR )
22 negeq 7919 . . . . . . . . . . . . 13  |-  ( x  =  -u y  ->  -u x  =  -u -u y )
2322oveq2d 5756 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  ( -u A  x.  -u x
)  =  ( -u A  x.  -u -u y
) )
2423eqeq1d 2124 . . . . . . . . . . 11  |-  ( x  =  -u y  ->  (
( -u A  x.  -u x
)  =  1  <->  ( -u A  x.  -u -u y
)  =  1 ) )
2524rspcev 2761 . . . . . . . . . 10  |-  ( (
-u y  e.  RR  /\  ( -u A  x.  -u -u y )  =  1 )  ->  E. x  e.  RR  ( -u A  x.  -u x )  =  1 )
2621, 25sylan 279 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  ( -u A  x.  -u -u y
)  =  1 )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
2720, 26sylbir 134 . . . . . . . 8  |-  ( ( y  e.  RR  /\  ( -u A  x.  y
)  =  1 )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
2827adantl 273 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( y  e.  RR  /\  ( -u A  x.  y )  =  1 ) )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
2915, 28rexlimddv 2529 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
30 recn 7717 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
31 recn 7717 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
32 mul2neg 8124 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( -u A  x.  -u x )  =  ( A  x.  x ) )
3330, 31, 32syl2an 285 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u A  x.  -u x )  =  ( A  x.  x ) )
3433eqeq1d 2124 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( -u A  x.  -u x )  =  1  <->  ( A  x.  x )  =  1 ) )
3534rexbidva 2409 . . . . . . 7  |-  ( A  e.  RR  ->  ( E. x  e.  RR  ( -u A  x.  -u x
)  =  1  <->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
3635adantr 272 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
( E. x  e.  RR  ( -u A  x.  -u x )  =  1  <->  E. x  e.  RR  ( A  x.  x
)  =  1 ) )
3729, 36mpbid 146 . . . . 5  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
3837ex 114 . . . 4  |-  ( A  e.  RR  ->  ( A  <  0  ->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
39 ltxrlt 7794 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
401, 39mpan 418 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
4140pm5.32i 447 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  <->  ( A  e.  RR  /\  0  <RR  A ) )
42 ax-precex 7694 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
43 simpr 109 . . . . . . . 8  |-  ( ( 0  <RR  x  /\  ( A  x.  x )  =  1 )  -> 
( A  x.  x
)  =  1 )
4443reximi 2504 . . . . . . 7  |-  ( E. x  e.  RR  (
0  <RR  x  /\  ( A  x.  x )  =  1 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
4542, 44syl 14 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
4641, 45sylbi 120 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
4746ex 114 . . . 4  |-  ( A  e.  RR  ->  (
0  <  A  ->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
4838, 47jaod 689 . . 3  |-  ( A  e.  RR  ->  (
( A  <  0  \/  0  <  A )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 ) )
493, 48sylbid 149 . 2  |-  ( A  e.  RR  ->  ( A #  0  ->  E. x  e.  RR  ( A  x.  x
)  =  1 ) )
5049imp 123 1  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   E.wrex 2392   class class class wbr 3897  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    <RR cltrr 7588    x. cmul 7589    < clt 7764   -ucneg 7898   # creap 8299
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-ltxr 7769  df-sub 7899  df-neg 7900  df-reap 8300
This theorem is referenced by:  rimul  8310  recexap  8377  rerecclap  8453
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