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Theorem rereceu 7792
Description: The reciprocal from axprecex 7783 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 7783 . . 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 2554 . . 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 2177 . . . . 5 |- ( ( ( A x. x ) = 1 /\ ( A x. y ) = 1 ) -> ( A x. x ) = ( A x. y ) )
6 axprecex 7783 . . . . . . 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 7763 . . . . . . . . . . . . 13 |- RR C_ CC
9 simpll 519 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. RR )
108, 9sseldi 3126 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. CC )
11 simprl 521 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
128, 11sseldi 3126 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
13 axmulcom 7774 . . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( A x. x ) = ( x x. A ) )
1410, 12, 13syl2anc 409 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( A x. x ) = ( x x. A ) )
15 simprr 522 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
168, 15sseldi 3126 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
17 axmulcom 7774 . . . . . . . . . . . 12 |- ( ( A e. CC /\ y e. CC ) -> ( A x. y ) = ( y x. A ) )
1810, 16, 17syl2anc 409 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( A x. y ) = ( y x. A ) )
1914, 18eqeq12d 2172 . . . . . . . . . 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 5825 . . . . . . . . 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 521 . . . . . . . . . . 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 3126 . . . . . . . . . 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 7776 . . . . . . . . . 10 |- ( ( x e. CC /\ A e. CC /\ z e. CC ) -> ( ( x x. A ) x. z ) = ( x x. ( A x. z ) ) )
2823, 24, 26, 27syl3anc 1220 . . . . . . . . 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 7776 . . . . . . . . . 10 |- ( ( y e. CC /\ A e. CC /\ z e. CC ) -> ( ( y x. A ) x. z ) = ( y x. ( A x. z ) ) )
3129, 24, 26, 30syl3anc 1220 . . . . . . . . 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 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 ) = ( ( 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 5826 . . . . . . . . . 10 |- ( ( A x. z ) = 1 -> ( x x. ( A x. z ) ) = ( x x. 1 ) )
3534ad2antll 483 . . . . . . . . 9 |- ( ( z e. RR /\ ( 0 <RR z /\ ( A x. z ) = 1 ) ) -> ( x x. ( A x. z ) ) = ( x x. 1 ) )
36 ax1rid 7780 . . . . . . . . . 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 2212 . . . . . . . 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 5826 . . . . . . . . . 10 |- ( ( A x. z ) = 1 -> ( y x. ( A x. z ) ) = ( y x. 1 ) )
4039ad2antll 483 . . . . . . . . 9 |- ( ( z e. RR /\ ( 0 <RR z /\ ( A x. z ) = 1 ) ) -> ( y x. ( A x. z ) ) = ( y x. 1 ) )
41 ax1rid 7780 . . . . . . . . . 10 |- ( y e. RR -> ( y x. 1 ) = y )
4241ad2antll 483 . . . . . . . . 9 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( y x. 1 ) = y )
4340, 42sylan9eqr 2212 . . . . . . . 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 2172 . . . . . . 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 2579 . . . . 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 2539 . . 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 5826 . . . . 5 |- ( x = y -> ( A x. x ) = ( A x. y ) )
5049eqeq1d 2166 . . . 4 |- ( x = y -> ( ( A x. x ) = 1 <-> ( A x. y ) = 1 ) )
5150rmo4 2905 . . 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 2669 . 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 414 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 1335   e. wcel 2128   A.wral 2435   E.wrex 2436   E!wreu 2437   E*wrmo 2438   class class class wbr 3965  (class class class)co 5818   CCcc 7713   RRcr 7714   0cc0 7715   1c1 7716   <RR cltrr 7719   x. cmul 7720
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4248  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-1o 6357  df-2o 6358  df-oadd 6361  df-omul 6362  df-er 6473  df-ec 6475  df-qs 6479  df-ni 7207  df-pli 7208  df-mi 7209  df-lti 7210  df-plpq 7247  df-mpq 7248  df-enq 7250  df-nqqs 7251  df-plqqs 7252  df-mqqs 7253  df-1nqqs 7254  df-rq 7255  df-ltnqqs 7256  df-enq0 7327  df-nq0 7328  df-0nq0 7329  df-plq0 7330  df-mq0 7331  df-inp 7369  df-i1p 7370  df-iplp 7371  df-imp 7372  df-iltp 7373  df-enr 7629  df-nr 7630  df-plr 7631  df-mr 7632  df-ltr 7633  df-0r 7634  df-1r 7635  df-m1r 7636  df-c 7721  df-0 7722  df-1 7723  df-r 7725  df-mul 7727  df-lt 7728
This theorem is referenced by:  recriota  7793
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