ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rereceu Unicode version
Theorem rereceu 8246
Description: The reciprocal from axprecex 8237 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 8237 . . 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 2647 . . 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 2258 . . . . 5 |- ( ( ( A x. x ) = 1 /\ ( A x. y ) = 1 ) -> ( A x. x ) = ( A x. y ) )
6 axprecex 8237 . . . . . . 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 8217 . . . . . . . . . . . . 13 |- RR C_ CC
9 simpll 531 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. RR )
108, 9sselid 3246 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> A e. CC )
11 simprl 535 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
128, 11sselid 3246 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
13 axmulcom 8228 . . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( A x. x ) = ( x x. A ) )
1410, 12, 13syl2anc 415 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( A x. x ) = ( x x. A ) )
15 simprr 537 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
168, 15sselid 3246 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
17 axmulcom 8228 . . . . . . . . . . . 12 |- ( ( A e. CC /\ y e. CC ) -> ( A x. y ) = ( y x. A ) )
1810, 16, 17syl2anc 415 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( A x. y ) = ( y x. A ) )
1914, 18eqeq12d 2253 . . . . . . . . . 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 6082 . . . . . . . . 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 535 . . . . . . . . . . 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 3246 . . . . . . . . . 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 8230 . . . . . . . . . 10 |- ( ( x e. CC /\ A e. CC /\ z e. CC ) -> ( ( x x. A ) x. z ) = ( x x. ( A x. z ) ) )
2823, 24, 26, 27syl3anc 1278 . . . . . . . . 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 8230 . . . . . . . . . 10 |- ( ( y e. CC /\ A e. CC /\ z e. CC ) -> ( ( y x. A ) x. z ) = ( y x. ( A x. z ) ) )
3129, 24, 26, 30syl3anc 1278 . . . . . . . . 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 2253 . . . . . . . 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 6083 . . . . . . . . . 10 |- ( ( A x. z ) = 1 -> ( x x. ( A x. z ) ) = ( x x. 1 ) )
3534ad2antll 495 . . . . . . . . 9 |- ( ( z e. RR /\ ( 0 <RR z /\ ( A x. z ) = 1 ) ) -> ( x x. ( A x. z ) ) = ( x x. 1 ) )
36 ax1rid 8234 . . . . . . . . . 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 2293 . . . . . . . 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 6083 . . . . . . . . . 10 |- ( ( A x. z ) = 1 -> ( y x. ( A x. z ) ) = ( y x. 1 ) )
4039ad2antll 495 . . . . . . . . 9 |- ( ( z e. RR /\ ( 0 <RR z /\ ( A x. z ) = 1 ) ) -> ( y x. ( A x. z ) ) = ( y x. 1 ) )
41 ax1rid 8234 . . . . . . . . . 10 |- ( y e. RR -> ( y x. 1 ) = y )
4241ad2antll 495 . . . . . . . . 9 |- ( ( ( A e. RR /\ 0 <RR A ) /\ ( x e. RR /\ y e. RR ) ) -> ( y x. 1 ) = y )
4340, 42sylan9eqr 2293 . . . . . . . 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 2253 . . . . . . 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 2673 . . . . 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 2632 . . 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 6083 . . . . 5 |- ( x = y -> ( A x. x ) = ( A x. y ) )
5049eqeq1d 2247 . . . 4 |- ( x = y -> ( ( A x. x ) = 1 <-> ( A x. y ) = 1 ) )
5150rmo4 3019 . . 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 2770 . 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 421 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 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   E*wrmo 2531   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   <RR cltrr 8173   x. cmul 8174
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-imp 7826  df-iltp 7827  df-enr 8083  df-nr 8084  df-plr 8085  df-mr 8086  df-ltr 8087  df-0r 8088  df-1r 8089  df-m1r 8090  df-c 8175  df-0 8176  df-1 8177  df-r 8179  df-mul 8181  df-lt 8182
This theorem is referenced by:  recriota  8247
  Copyright terms: Public domain W3C validator