Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xreceu Structured version   Unicode version

Theorem xreceu 26097
Description: Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
Assertion
Ref Expression
xreceu  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
Distinct variable groups:    x, A    x, B

Proof of Theorem xreceu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ressxr 9427 . . . 4  |-  RR  C_  RR*
2 xrecex 26095 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B xe y )  =  1 )
323adant1 1006 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B xe y )  =  1 )
4 ssrexv 3417 . . . 4  |-  ( RR  C_  RR*  ->  ( E. y  e.  RR  ( B xe y )  =  1  ->  E. y  e.  RR*  ( B xe y )  =  1 ) )
51, 3, 4mpsyl 63 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR*  ( B xe y )  =  1 )
6 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  y  e.  RR* )
7 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  A  e.  RR* )
86, 7xmulcld 11265 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( y xe A )  e. 
RR* )
9 oveq1 6098 . . . . . . . 8  |-  ( ( B xe y )  =  1  -> 
( ( B xe y ) xe A )  =  ( 1 xe A ) )
109ad2antll 728 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( ( B xe y ) xe A )  =  ( 1 xe A ) )
11 simplr 754 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  B  e.  RR )
1211rexrd 9433 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  B  e.  RR* )
13 xmulass 11250 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  y  e.  RR*  /\  A  e. 
RR* )  ->  (
( B xe y ) xe A )  =  ( B xe ( y xe A ) ) )
1412, 6, 7, 13syl3anc 1218 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( ( B xe y ) xe A )  =  ( B xe ( y xe A ) ) )
15 xmulid2 11243 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( 1 xe A )  =  A )
167, 15syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( 1 xe A )  =  A )
1710, 14, 163eqtr3d 2483 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( B xe ( y xe A ) )  =  A )
18 oveq2 6099 . . . . . . . 8  |-  ( x  =  ( y xe A )  -> 
( B xe x )  =  ( B xe ( y xe A ) ) )
1918eqeq1d 2451 . . . . . . 7  |-  ( x  =  ( y xe A )  -> 
( ( B xe x )  =  A  <->  ( B xe ( y xe A ) )  =  A ) )
2019rspcev 3073 . . . . . 6  |-  ( ( ( y xe A )  e.  RR*  /\  ( B xe ( y xe A ) )  =  A )  ->  E. x  e.  RR*  ( B xe x )  =  A )
218, 17, 20syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  E. x  e.  RR*  ( B xe x )  =  A )
2221rexlimdvaa 2842 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( E. y  e.  RR*  ( B xe y )  =  1  ->  E. x  e.  RR*  ( B xe x )  =  A ) )
23223adant3 1008 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( E. y  e.  RR*  ( B xe y )  =  1  ->  E. x  e.  RR*  ( B xe x )  =  A ) )
245, 23mpd 15 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. x  e.  RR*  ( B xe x )  =  A )
25 eqtr3 2462 . . . . . . 7  |-  ( ( ( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  ( B xe x )  =  ( B xe y ) )
26 simp1 988 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  x  e.  RR* )
27 simp2 989 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
y  e.  RR* )
28 simp3l 1016 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  e.  RR )
29 simp3r 1017 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  =/=  0 )
3026, 27, 28, 29xmulcand 26096 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( ( B xe x )  =  ( B xe y )  <->  x  =  y ) )
3125, 30syl5ib 219 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( ( ( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  x  =  y )
)
32313expa 1187 . . . . 5  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( (
( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  x  =  y ) )
3332expcom 435 . . . 4  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( (
( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  x  =  y ) ) )
34333adant1 1006 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( ( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  x  =  y )
) )
3534ralrimivv 2807 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  A. x  e.  RR*  A. y  e. 
RR*  ( ( ( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  x  =  y ) )
36 oveq2 6099 . . . 4  |-  ( x  =  y  ->  ( B xe x )  =  ( B xe y ) )
3736eqeq1d 2451 . . 3  |-  ( x  =  y  ->  (
( B xe x )  =  A  <-> 
( B xe y )  =  A ) )
3837reu4 3153 . 2  |-  ( E! x  e.  RR*  ( B xe x )  =  A  <->  ( E. x  e.  RR*  ( B xe x )  =  A  /\  A. x  e.  RR*  A. y  e.  RR*  ( ( ( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  x  =  y ) ) )
3924, 35, 38sylanbrc 664 1  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   E!wreu 2717    C_ wss 3328  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283   RR*cxr 9417   xecxmu 11088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-xneg 11089  df-xmul 11091
This theorem is referenced by:  xdivcld  26098  xdivmul  26100  rexdiv  26101  xrmulc1cn  26360
  Copyright terms: Public domain W3C validator