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Theorem rereceu 6961
Description: The reciprocal from axprecex 6952 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
Assertion
Ref Expression
rereceu  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E! x  e.  RR  ( A  x.  x
)  =  1 )
Distinct variable group:    x, A

Proof of Theorem rereceu
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axprecex 6952 . . 3  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
2 simpr 103 . . . 4  |-  ( ( 0  <RR  x  /\  ( A  x.  x )  =  1 )  -> 
( A  x.  x
)  =  1 )
32reximi 2416 . . 3  |-  ( E. x  e.  RR  (
0  <RR  x  /\  ( A  x.  x )  =  1 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
41, 3syl 14 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
5 eqtr3 2059 . . . . 5  |-  ( ( ( A  x.  x
)  =  1  /\  ( A  x.  y
)  =  1 )  ->  ( A  x.  x )  =  ( A  x.  y ) )
6 axprecex 6952 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. z  e.  RR  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )
76adantr 261 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  E. z  e.  RR  ( 0  <RR  z  /\  ( A  x.  z )  =  1 ) )
8 axresscn 6934 . . . . . . . . . . . . 13  |-  RR  C_  CC
9 simpll 481 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  RR )
108, 9sseldi 2943 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  CC )
11 simprl 483 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
128, 11sseldi 2943 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  CC )
13 axmulcom 6943 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  =  ( x  x.  A ) )
1410, 12, 13syl2anc 391 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  x.  x )  =  ( x  x.  A ) )
15 simprr 484 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  RR )
168, 15sseldi 2943 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  CC )
17 axmulcom 6943 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  =  ( y  x.  A ) )
1810, 16, 17syl2anc 391 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  x.  y )  =  ( y  x.  A ) )
1914, 18eqeq12d 2054 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( ( A  x.  x )  =  ( A  x.  y )  <->  ( x  x.  A )  =  ( y  x.  A ) ) )
2019adantr 261 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( A  x.  x
)  =  ( A  x.  y )  <->  ( x  x.  A )  =  ( y  x.  A ) ) )
21 oveq1 5519 . . . . . . . . 9  |-  ( ( x  x.  A )  =  ( y  x.  A )  ->  (
( x  x.  A
)  x.  z )  =  ( ( y  x.  A )  x.  z ) )
2220, 21syl6bi 152 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( A  x.  x
)  =  ( A  x.  y )  -> 
( ( x  x.  A )  x.  z
)  =  ( ( y  x.  A )  x.  z ) ) )
2312adantr 261 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  x  e.  CC )
2410adantr 261 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  A  e.  CC )
25 simprl 483 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  z  e.  RR )
268, 25sseldi 2943 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  z  e.  CC )
27 axmulass 6945 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  A
)  x.  z )  =  ( x  x.  ( A  x.  z
) ) )
2823, 24, 26, 27syl3anc 1135 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( x  x.  A
)  x.  z )  =  ( x  x.  ( A  x.  z
) ) )
2916adantr 261 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  y  e.  CC )
30 axmulass 6945 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  z  e.  CC )  ->  (
( y  x.  A
)  x.  z )  =  ( y  x.  ( A  x.  z
) ) )
3129, 24, 26, 30syl3anc 1135 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( y  x.  A
)  x.  z )  =  ( y  x.  ( A  x.  z
) ) )
3228, 31eqeq12d 2054 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( ( x  x.  A )  x.  z
)  =  ( ( y  x.  A )  x.  z )  <->  ( x  x.  ( A  x.  z
) )  =  ( y  x.  ( A  x.  z ) ) ) )
3322, 32sylibd 138 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( A  x.  x
)  =  ( A  x.  y )  -> 
( x  x.  ( A  x.  z )
)  =  ( y  x.  ( A  x.  z ) ) ) )
34 oveq2 5520 . . . . . . . . . 10  |-  ( ( A  x.  z )  =  1  ->  (
x  x.  ( A  x.  z ) )  =  ( x  x.  1 ) )
3534ad2antll 460 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )  ->  ( x  x.  ( A  x.  z
) )  =  ( x  x.  1 ) )
36 ax1rid 6949 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (
x  x.  1 )  =  x )
3711, 36syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  1 )  =  x )
3835, 37sylan9eqr 2094 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
x  x.  ( A  x.  z ) )  =  x )
39 oveq2 5520 . . . . . . . . . 10  |-  ( ( A  x.  z )  =  1  ->  (
y  x.  ( A  x.  z ) )  =  ( y  x.  1 ) )
4039ad2antll 460 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )  ->  ( y  x.  ( A  x.  z
) )  =  ( y  x.  1 ) )
41 ax1rid 6949 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
4241ad2antll 460 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( y  x.  1 )  =  y )
4340, 42sylan9eqr 2094 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
y  x.  ( A  x.  z ) )  =  y )
4438, 43eqeq12d 2054 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( x  x.  ( A  x.  z )
)  =  ( y  x.  ( A  x.  z ) )  <->  x  =  y ) )
4533, 44sylibd 138 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) ) )  ->  (
( A  x.  x
)  =  ( A  x.  y )  ->  x  =  y )
)
467, 45rexlimddv 2437 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( ( A  x.  x )  =  ( A  x.  y )  ->  x  =  y ) )
475, 46syl5 28 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( (
( A  x.  x
)  =  1  /\  ( A  x.  y
)  =  1 )  ->  x  =  y ) )
4847ralrimivva 2401 . . 3  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  A. x  e.  RR  A. y  e.  RR  (
( ( A  x.  x )  =  1  /\  ( A  x.  y )  =  1 )  ->  x  =  y ) )
49 oveq2 5520 . . . . 5  |-  ( x  =  y  ->  ( A  x.  x )  =  ( A  x.  y ) )
5049eqeq1d 2048 . . . 4  |-  ( x  =  y  ->  (
( A  x.  x
)  =  1  <->  ( A  x.  y )  =  1 ) )
5150rmo4 2734 . . 3  |-  ( E* x  e.  RR  ( A  x.  x )  =  1  <->  A. x  e.  RR  A. y  e.  RR  ( ( ( A  x.  x )  =  1  /\  ( A  x.  y )  =  1 )  ->  x  =  y )
)
5248, 51sylibr 137 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E* x  e.  RR  ( A  x.  x
)  =  1 )
53 reu5 2522 . 2  |-  ( E! x  e.  RR  ( A  x.  x )  =  1  <->  ( E. x  e.  RR  ( A  x.  x )  =  1  /\  E* x  e.  RR  ( A  x.  x )  =  1 ) )
544, 52, 53sylanbrc 394 1  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E! x  e.  RR  ( A  x.  x
)  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   E.wrex 2307   E!wreu 2308   E*wrmo 2309   class class class wbr 3764  (class class class)co 5512   CCcc 6885   RRcr 6886   0cc0 6887   1c1 6888    <RR cltrr 6891    x. cmul 6892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565  df-iltp 6566  df-enr 6809  df-nr 6810  df-plr 6811  df-mr 6812  df-ltr 6813  df-0r 6814  df-1r 6815  df-m1r 6816  df-c 6893  df-0 6894  df-1 6895  df-r 6897  df-mul 6899  df-lt 6900
This theorem is referenced by:  recriota  6962
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