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Theorem rereceu 7665
Description: The reciprocal from axprecex 7656 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 7656 . . 3  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
2 simpr 109 . . . 4  |-  ( ( 0  <RR  x  /\  ( A  x.  x )  =  1 )  -> 
( A  x.  x
)  =  1 )
32reximi 2506 . . 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 2137 . . . . 5  |-  ( ( ( A  x.  x
)  =  1  /\  ( A  x.  y
)  =  1 )  ->  ( A  x.  x )  =  ( A  x.  y ) )
6 axprecex 7656 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. z  e.  RR  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )
76adantr 274 . . . . . 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 7636 . . . . . . . . . . . . 13  |-  RR  C_  CC
9 simpll 503 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  RR )
108, 9sseldi 3065 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  CC )
11 simprl 505 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
128, 11sseldi 3065 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  CC )
13 axmulcom 7647 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  =  ( x  x.  A ) )
1410, 12, 13syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  x.  x )  =  ( x  x.  A ) )
15 simprr 506 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  RR )
168, 15sseldi 3065 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  CC )
17 axmulcom 7647 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  =  ( y  x.  A ) )
1810, 16, 17syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  x.  y )  =  ( y  x.  A ) )
1914, 18eqeq12d 2132 . . . . . . . . . 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 274 . . . . . . . . 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 5749 . . . . . . . . 9  |-  ( ( x  x.  A )  =  ( y  x.  A )  ->  (
( x  x.  A
)  x.  z )  =  ( ( y  x.  A )  x.  z ) )
2220, 21syl6bi 162 . . . . . . . 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 274 . . . . . . . . . 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 274 . . . . . . . . . 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 505 . . . . . . . . . . 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 3065 . . . . . . . . . 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 7649 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  A  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  A
)  x.  z )  =  ( x  x.  ( A  x.  z
) ) )
2823, 24, 26, 27syl3anc 1201 . . . . . . . . 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 274 . . . . . . . . . 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 7649 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  z  e.  CC )  ->  (
( y  x.  A
)  x.  z )  =  ( y  x.  ( A  x.  z
) ) )
3129, 24, 26, 30syl3anc 1201 . . . . . . . . 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 2132 . . . . . . . 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 148 . . . . . . 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 5750 . . . . . . . . . 10  |-  ( ( A  x.  z )  =  1  ->  (
x  x.  ( A  x.  z ) )  =  ( x  x.  1 ) )
3534ad2antll 482 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )  ->  ( x  x.  ( A  x.  z
) )  =  ( x  x.  1 ) )
36 ax1rid 7653 . . . . . . . . . 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 2172 . . . . . . . 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 5750 . . . . . . . . . 10  |-  ( ( A  x.  z )  =  1  ->  (
y  x.  ( A  x.  z ) )  =  ( y  x.  1 ) )
4039ad2antll 482 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( 0  <RR  z  /\  ( A  x.  z
)  =  1 ) )  ->  ( y  x.  ( A  x.  z
) )  =  ( y  x.  1 ) )
41 ax1rid 7653 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
4241ad2antll 482 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <RR  A )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( y  x.  1 )  =  y )
4340, 42sylan9eqr 2172 . . . . . . . 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 2132 . . . . . . 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 148 . . . . . 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 2531 . . . . 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 2491 . . 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 5750 . . . . 5  |-  ( x  =  y  ->  ( A  x.  x )  =  ( A  x.  y ) )
5049eqeq1d 2126 . . . 4  |-  ( x  =  y  ->  (
( A  x.  x
)  =  1  <->  ( A  x.  y )  =  1 ) )
5150rmo4 2850 . . 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 133 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E* x  e.  RR  ( A  x.  x
)  =  1 )
53 reu5 2620 . 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 413 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   E!wreu 2395   E*wrmo 2396   class class class wbr 3899  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    <RR cltrr 7592    x. cmul 7593
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-imp 7245  df-iltp 7246  df-enr 7502  df-nr 7503  df-plr 7504  df-mr 7505  df-ltr 7506  df-0r 7507  df-1r 7508  df-m1r 7509  df-c 7594  df-0 7595  df-1 7596  df-r 7598  df-mul 7600  df-lt 7601
This theorem is referenced by:  recriota  7666
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