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Theorem recexre 7643
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 7085 . . . 4  |-  0  e.  RR
2 reapval 7641 . . . 4  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  ( A  <  0  \/  0  < 
A ) ) )
31, 2mpan2 409 . . 3  |-  ( A  e.  RR  ->  ( A #  0 
<->  ( A  <  0  \/  0  <  A ) ) )
4 lt0neg1 7537 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
5 renegcl 7335 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 ltxrlt 7144 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  -u A  e.  RR )  ->  ( 0  <  -u A  <->  0  <RR  -u A
) )
71, 5, 6sylancr 399 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
0  <  -u A  <->  0  <RR  -u A ) )
84, 7bitrd 181 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <RR  -u A ) )
98pm5.32i 435 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  0 )  <->  ( A  e.  RR  /\  0  <RR  -u A ) )
10 ax-precex 7052 . . . . . . . . . 10  |-  ( (
-u A  e.  RR  /\  0  <RR  -u A )  ->  E. y  e.  RR  ( 0  <RR  y  /\  ( -u A  x.  y
)  =  1 ) )
11 simpr 107 . . . . . . . . . . 11  |-  ( ( 0  <RR  y  /\  ( -u A  x.  y )  =  1 )  -> 
( -u A  x.  y
)  =  1 )
1211reximi 2433 . . . . . . . . . 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 271 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <RR  -u A )  ->  E. y  e.  RR  ( -u A  x.  y
)  =  1 )
159, 14sylbi 118 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  E. y  e.  RR  ( -u A  x.  y
)  =  1 )
16 recn 7072 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  y  e.  CC )
1716negnegd 7376 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  -u -u y  =  y )
1817oveq2d 5556 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  ( -u A  x.  -u -u y
)  =  ( -u A  x.  y )
)
1918eqeq1d 2064 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
( -u A  x.  -u -u y
)  =  1  <->  ( -u A  x.  y )  =  1 ) )
2019pm5.32i 435 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  ( -u A  x.  -u -u y
)  =  1 )  <-> 
( y  e.  RR  /\  ( -u A  x.  y )  =  1 ) )
21 renegcl 7335 . . . . . . . . . 10  |-  ( y  e.  RR  ->  -u y  e.  RR )
22 negeq 7267 . . . . . . . . . . . . 13  |-  ( x  =  -u y  ->  -u x  =  -u -u y )
2322oveq2d 5556 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  ( -u A  x.  -u x
)  =  ( -u A  x.  -u -u y
) )
2423eqeq1d 2064 . . . . . . . . . . 11  |-  ( x  =  -u y  ->  (
( -u A  x.  -u x
)  =  1  <->  ( -u A  x.  -u -u y
)  =  1 ) )
2524rspcev 2673 . . . . . . . . . 10  |-  ( (
-u y  e.  RR  /\  ( -u A  x.  -u -u y )  =  1 )  ->  E. x  e.  RR  ( -u A  x.  -u x )  =  1 )
2621, 25sylan 271 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  ( -u A  x.  -u -u y
)  =  1 )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
2720, 26sylbir 129 . . . . . . . 8  |-  ( ( y  e.  RR  /\  ( -u A  x.  y
)  =  1 )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
2827adantl 266 . . . . . . 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 2454 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  E. x  e.  RR  ( -u A  x.  -u x
)  =  1 )
30 recn 7072 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
31 recn 7072 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
32 mul2neg 7467 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( -u A  x.  -u x )  =  ( A  x.  x ) )
3330, 31, 32syl2an 277 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u A  x.  -u x )  =  ( A  x.  x ) )
3433eqeq1d 2064 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( -u A  x.  -u x )  =  1  <->  ( A  x.  x )  =  1 ) )
3534rexbidva 2340 . . . . . . 7  |-  ( A  e.  RR  ->  ( E. x  e.  RR  ( -u A  x.  -u x
)  =  1  <->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
3635adantr 265 . . . . . 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 139 . . . . 5  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
3837ex 112 . . . 4  |-  ( A  e.  RR  ->  ( A  <  0  ->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
39 ltxrlt 7144 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
401, 39mpan 408 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
4140pm5.32i 435 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  <->  ( A  e.  RR  /\  0  <RR  A ) )
42 ax-precex 7052 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
43 simpr 107 . . . . . . . 8  |-  ( ( 0  <RR  x  /\  ( A  x.  x )  =  1 )  -> 
( A  x.  x
)  =  1 )
4443reximi 2433 . . . . . . 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 118 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
4746ex 112 . . . 4  |-  ( A  e.  RR  ->  (
0  <  A  ->  E. x  e.  RR  ( A  x.  x )  =  1 ) )
4838, 47jaod 647 . . 3  |-  ( A  e.  RR  ->  (
( A  <  0  \/  0  <  A )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 ) )
493, 48sylbid 143 . 2  |-  ( A  e.  RR  ->  ( A #  0  ->  E. x  e.  RR  ( A  x.  x
)  =  1 ) )
5049imp 119 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 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    <RR cltrr 6951    x. cmul 6952    < clt 7119   -ucneg 7246   # creap 7639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-ltxr 7124  df-sub 7247  df-neg 7248  df-reap 7640
This theorem is referenced by:  rimul  7650  recexap  7708  rerecclap  7781
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