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Theorem rereceu 7106
Description: The reciprocal from axprecex 7097 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 7097 . . 3  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
2 simpr 108 . . . 4  |-  ( ( 0  <RR  x  /\  ( A  x.  x )  =  1 )  -> 
( A  x.  x
)  =  1 )
32reximi 2459 . . 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 2101 . . . . 5  |-  ( ( ( A  x.  x
)  =  1  /\  ( A  x.  y
)  =  1 )  ->  ( A  x.  x )  =  ( A  x.  y ) )
6 axprecex 7097 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. z  e.  RR  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )
76adantr 270 . . . . . 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 7079 . . . . . . . . . . . . 13  |-  RR  C_  CC
9 simpll 496 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  RR )
108, 9sseldi 2998 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  CC )
11 simprl 498 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
128, 11sseldi 2998 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  CC )
13 axmulcom 7088 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  =  ( x  x.  A ) )
1410, 12, 13syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  x.  x )  =  ( x  x.  A ) )
15 simprr 499 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  RR )
168, 15sseldi 2998 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  CC )
17 axmulcom 7088 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  =  ( y  x.  A ) )
1810, 16, 17syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  x.  y )  =  ( y  x.  A ) )
1914, 18eqeq12d 2096 . . . . . . . . . 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 270 . . . . . . . . 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 5544 . . . . . . . . 9  |-  ( ( x  x.  A )  =  ( y  x.  A )  ->  (
( x  x.  A
)  x.  z )  =  ( ( y  x.  A )  x.  z ) )
2220, 21syl6bi 161 . . . . . . . 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 270 . . . . . . . . . 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 270 . . . . . . . . . 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 498 . . . . . . . . . . 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 2998 . . . . . . . . . 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 7090 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  A
)  x.  z )  =  ( x  x.  ( A  x.  z
) ) )
2823, 24, 26, 27syl3anc 1170 . . . . . . . . 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 270 . . . . . . . . . 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 7090 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  z  e.  CC )  ->  (
( y  x.  A
)  x.  z )  =  ( y  x.  ( A  x.  z
) ) )
3129, 24, 26, 30syl3anc 1170 . . . . . . . . 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 2096 . . . . . . . 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 147 . . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( A  x.  z )  =  1  ->  (
x  x.  ( A  x.  z ) )  =  ( x  x.  1 ) )
3534ad2antll 475 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )  ->  ( x  x.  ( A  x.  z
) )  =  ( x  x.  1 ) )
36 ax1rid 7094 . . . . . . . . . 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 2136 . . . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( A  x.  z )  =  1  ->  (
y  x.  ( A  x.  z ) )  =  ( y  x.  1 ) )
4039ad2antll 475 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )  ->  ( y  x.  ( A  x.  z
) )  =  ( y  x.  1 ) )
41 ax1rid 7094 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
4241ad2antll 475 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( y  x.  1 )  =  y )
4340, 42sylan9eqr 2136 . . . . . . . 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 2096 . . . . . . 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 147 . . . . . 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 2482 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( ( A  x.  x )  =  ( A  x.  y )  ->  x  =  y ) )
475, 46syl5 32 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( (
( A  x.  x
)  =  1  /\  ( A  x.  y
)  =  1 )  ->  x  =  y ) )
4847ralrimivva 2444 . . 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 5545 . . . . 5  |-  ( x  =  y  ->  ( A  x.  x )  =  ( A  x.  y ) )
5049eqeq1d 2090 . . . 4  |-  ( x  =  y  ->  (
( A  x.  x
)  =  1  <->  ( A  x.  y )  =  1 ) )
5150rmo4 2786 . . 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 132 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E* x  e.  RR  ( A  x.  x
)  =  1 )
53 reu5 2567 . 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 408 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   E!wreu 2351   E*wrmo 2352   class class class wbr 3787  (class class class)co 5537   CCcc 7030   RRcr 7031   0cc0 7032   1c1 7033    <RR cltrr 7036    x. cmul 7037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-i1p 6708  df-iplp 6709  df-imp 6710  df-iltp 6711  df-enr 6954  df-nr 6955  df-plr 6956  df-mr 6957  df-ltr 6958  df-0r 6959  df-1r 6960  df-m1r 6961  df-c 7038  df-0 7039  df-1 7040  df-r 7042  df-mul 7044  df-lt 7045
This theorem is referenced by:  recriota  7107
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