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Theorem rereceu 7902
Description: The reciprocal from axprecex 7893 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 7893 . . 3 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
2 simpr 110 . . . 4 |- ( ( 0 <RR x /\ ( A x. x ) = 1 ) -> ( A x. x ) = 1 )
32reximi 2584 . . 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 2207 . . . . 5 |- ( ( ( A x. x ) = 1 /\ ( A x. y ) = 1 ) -> ( A x. x ) = ( A x. y ) )
6 axprecex 7893 . . . . . . 7 |- ( ( A e. RR /\ 0 <RR A ) -> E. z e. RR ( 0 <RR z /\ ( A x. z ) = 1 ) )
76adantr 276 . . . . . 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 7873 . . . . . . . . . . . . 13 |- RR C_ CC
9 simpll 527 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. RR )
108, 9sselid 3165 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. CC )
11 simprl 529 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
128, 11sselid 3165 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
13 axmulcom 7884 . . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( A x. x ) = ( x x. A ) )
1410, 12, 13syl2anc 411 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( A x. x ) = ( x x. A ) )
15 simprr 531 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
168, 15sselid 3165 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
17 axmulcom 7884 . . . . . . . . . . . 12 |- ( ( A e. CC /\ y e. CC ) -> ( A x. y ) = ( y x. A ) )
1810, 16, 17syl2anc 411 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( A x. y ) = ( y x. A ) )
1914, 18eqeq12d 2202 . . . . . . . . . 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 276 . . . . . . . . 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 5895 . . . . . . . . 9 |- ( ( x x. A ) = ( y x. A ) -> ( ( x x. A ) x. z ) = ( ( y x. A ) x. z ) )
2220, 21biimtrdi 163 . . . . . . . 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 276 . . . . . . . . . 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 276 . . . . . . . . . 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 529 . . . . . . . . . . 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, 25sselid 3165 . . . . . . . . . 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 7886 . . . . . . . . . 10 |- ( ( x e. CC /\ A e. CC /\ z e. CC ) -> ( ( x x. A ) x. z ) = ( x x. ( A x. z ) ) )
2823, 24, 26, 27syl3anc 1248 . . . . . . . . 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 276 . . . . . . . . . 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 7886 . . . . . . . . . 10 |- ( ( y e. CC /\ A e. CC /\ z e. CC ) -> ( ( y x. A ) x. z ) = ( y x. ( A x. z ) ) )
3129, 24, 26, 30syl3anc 1248 . . . . . . . . 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 2202 . . . . . . . 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 149 . . . . . . 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 5896 . . . . . . . . . 10 |- ( ( A x. z ) = 1 -> ( x x. ( A x. z ) ) = ( x x. 1 ) )
3534ad2antll 491 . . . . . . . . 9 |- ( ( z e. RR /\ ( 0 <RR z /\ ( A x. z ) = 1 ) ) -> ( x x. ( A x. z ) ) = ( x x. 1 ) )
36 ax1rid 7890 . . . . . . . . . 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 2242 . . . . . . . 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 5896 . . . . . . . . . 10 |- ( ( A x. z ) = 1 -> ( y x. ( A x. z ) ) = ( y x. 1 ) )
4039ad2antll 491 . . . . . . . . 9 |- ( ( z e. RR /\ ( 0 <RR z /\ ( A x. z ) = 1 ) ) -> ( y x. ( A x. z ) ) = ( y x. 1 ) )
41 ax1rid 7890 . . . . . . . . . 10 |- ( y e. RR -> ( y x. 1 ) = y )
4241ad2antll 491 . . . . . . . . 9 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( y x. 1 ) = y )
4340, 42sylan9eqr 2242 . . . . . . . 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 2202 . . . . . . 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 149 . . . . . 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 2609 . . . . 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 2569 . . 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 5896 . . . . 5 |- ( x = y -> ( A x. x ) = ( A x. y ) )
5049eqeq1d 2196 . . . 4 |- ( x = y -> ( ( A x. x ) = 1 <-> ( A x. y ) = 1 ) )
5150rmo4 2942 . . 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 134 . 2 |- ( ( A e. RR /\ 0 <RR A ) -> E* x e. RR ( A x. x ) = 1 )
53 reu5 2700 . 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 417 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 104   <-> wb 105   = wceq 1363   e. wcel 2158   A.wral 2465   E.wrex 2466   E!wreu 2467   E*wrmo 2468   class class class wbr 4015  (class class class)co 5888   CCcc 7823   RRcr 7824   0cc0 7825   1c1 7826   <RR cltrr 7829   x. cmul 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479  df-i1p 7480  df-iplp 7481  df-imp 7482  df-iltp 7483  df-enr 7739  df-nr 7740  df-plr 7741  df-mr 7742  df-ltr 7743  df-0r 7744  df-1r 7745  df-m1r 7746  df-c 7831  df-0 7832  df-1 7833  df-r 7835  df-mul 7837  df-lt 7838
This theorem is referenced by:  recriota  7903
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