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Theorem rereceu 8169
Description: The reciprocal from axprecex 8160 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 8160 . . 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 2630 . . 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 2251 . . . . 5 |- ( ( ( A x. x ) = 1 /\ ( A x. y ) = 1 ) -> ( A x. x ) = ( A x. y ) )
6 axprecex 8160 . . . . . . 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 8140 . . . . . . . . . . . . 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 3226 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. CC )
11 simprl 531 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
128, 11sselid 3226 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
13 axmulcom 8151 . . . . . . . . . . . 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 533 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
168, 15sselid 3226 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
17 axmulcom 8151 . . . . . . . . . . . 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 2246 . . . . . . . . . 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 6035 . . . . . . . . 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 531 . . . . . . . . . . 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 3226 . . . . . . . . . 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 8153 . . . . . . . . . 10 |- ( ( x e. CC /\ A e. CC /\ z e. CC ) -> ( ( x x. A ) x. z ) = ( x x. ( A x. z ) ) )
2823, 24, 26, 27syl3anc 1274 . . . . . . . . 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 8153 . . . . . . . . . 10 |- ( ( y e. CC /\ A e. CC /\ z e. CC ) -> ( ( y x. A ) x. z ) = ( y x. ( A x. z ) ) )
3129, 24, 26, 30syl3anc 1274 . . . . . . . . 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 2246 . . . . . . . 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 6036 . . . . . . . . . 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 8157 . . . . . . . . . 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 2286 . . . . . . . 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 6036 . . . . . . . . . 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 8157 . . . . . . . . . 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 2286 . . . . . . . 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 2246 . . . . . . 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 2656 . . . . 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 2615 . . 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 6036 . . . . 5 |- ( x = y -> ( A x. x ) = ( A x. y ) )
5049eqeq1d 2240 . . . 4 |- ( x = y -> ( ( A x. x ) = 1 <-> ( A x. y ) = 1 ) )
5150rmo4 3000 . . 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 2752 . 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 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   E!wreu 2513   E*wrmo 2514   class class class wbr 4093  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   <RR cltrr 8096   x. cmul 8097
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-i1p 7747  df-iplp 7748  df-imp 7749  df-iltp 7750  df-enr 8006  df-nr 8007  df-plr 8008  df-mr 8009  df-ltr 8010  df-0r 8011  df-1r 8012  df-m1r 8013  df-c 8098  df-0 8099  df-1 8100  df-r 8102  df-mul 8104  df-lt 8105
This theorem is referenced by:  recriota  8170
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