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Theorem xreceu 27778
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 9654 . . . 4  |-  RR  C_  RR*
2 xrecex 27776 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B xe y )  =  1 )
323adant1 1014 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B xe y )  =  1 )
4 ssrexv 3561 . . . 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 756 . . . . . . 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 11519 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( y xe A )  e. 
RR* )
9 oveq1 6303 . . . . . . . 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 755 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  B  e.  RR )
1211rexrd 9660 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  B  e.  RR* )
13 xmulass 11504 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  y  e.  RR*  /\  A  e. 
RR* )  ->  (
( B xe y ) xe A )  =  ( B xe ( y xe A ) ) )
1412, 6, 7, 13syl3anc 1228 . . . . . . 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 11497 . . . . . . . 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 2506 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B xe y )  =  1 ) )  ->  ( B xe ( y xe A ) )  =  A )
18 oveq2 6304 . . . . . . . 8  |-  ( x  =  ( y xe A )  -> 
( B xe x )  =  ( B xe ( y xe A ) ) )
1918eqeq1d 2459 . . . . . . 7  |-  ( x  =  ( y xe A )  -> 
( ( B xe x )  =  A  <->  ( B xe ( y xe A ) )  =  A ) )
2019rspcev 3210 . . . . . 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 2950 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( E. y  e.  RR*  ( B xe y )  =  1  ->  E. x  e.  RR*  ( B xe x )  =  A ) )
23223adant3 1016 . . 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 2485 . . . . . . 7  |-  ( ( ( B xe x )  =  A  /\  ( B xe y )  =  A )  ->  ( B xe x )  =  ( B xe y ) )
26 simp1 996 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  x  e.  RR* )
27 simp2 997 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
y  e.  RR* )
28 simp3l 1024 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  e.  RR )
29 simp3r 1025 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  =/=  0 )
3026, 27, 28, 29xmulcand 27777 . . . . . . 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 1196 . . . . 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 1014 . . 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 2877 . 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 6304 . . . 4  |-  ( x  =  y  ->  ( B xe x )  =  ( B xe y ) )
3736eqeq1d 2459 . . 3  |-  ( x  =  y  ->  (
( B xe x )  =  A  <-> 
( B xe y )  =  A ) )
3837reu4 3293 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   E!wreu 2809    C_ wss 3471  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510   RR*cxr 9644   xecxmu 11342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-xneg 11343  df-xmul 11345
This theorem is referenced by:  xdivcld  27779  xdivmul  27781  rexdiv  27782  xrmulc1cn  28073
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