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Theorem xreceu 24008
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 x e 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 9064 . . . 4  |-  RR  C_  RR*
2 xrecex 24006 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B x e y )  =  1 )
323adant1 975 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B x e y )  =  1 )
4 ssrexv 3353 . . . 4  |-  ( RR  C_  RR*  ->  ( E. y  e.  RR  ( B x e y )  =  1  ->  E. y  e.  RR*  ( B x e y )  =  1 ) )
51, 3, 4mpsyl 61 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR*  ( B x e y )  =  1 )
6 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
y  e.  RR* )
7 simpll 731 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  A  e.  RR* )
86, 7xmulcld 10815 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( y x e A )  e.  RR* )
9 oveq1 6029 . . . . . . . 8  |-  ( ( B x e y )  =  1  -> 
( ( B x e y ) x e A )  =  ( 1 x e A ) )
109ad2antll 710 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( ( B x e y ) x e A )  =  ( 1 x e A ) )
11 simplr 732 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  B  e.  RR )
1211rexrd 9069 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  B  e.  RR* )
13 xmulass 10800 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  y  e.  RR*  /\  A  e. 
RR* )  ->  (
( B x e y ) x e A )  =  ( B x e ( y x e A ) ) )
1412, 6, 7, 13syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( ( B x e y ) x e A )  =  ( B x e ( y x e A ) ) )
15 xmulid2 10793 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( 1 x e A )  =  A )
167, 15syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( 1 x e A )  =  A )
1710, 14, 163eqtr3d 2429 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( B x e ( y x e A ) )  =  A )
18 oveq2 6030 . . . . . . . 8  |-  ( x  =  ( y x e A )  -> 
( B x e x )  =  ( B x e ( y x e A ) ) )
1918eqeq1d 2397 . . . . . . 7  |-  ( x  =  ( y x e A )  -> 
( ( B x e x )  =  A  <->  ( B x e ( y x e A ) )  =  A ) )
2019rspcev 2997 . . . . . 6  |-  ( ( ( y x e A )  e.  RR*  /\  ( B x e ( y x e A ) )  =  A )  ->  E. x  e.  RR*  ( B x e x )  =  A )
218, 17, 20syl2anc 643 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  E. x  e.  RR*  ( B x e x )  =  A )
2221rexlimdvaa 2776 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( E. y  e.  RR*  ( B x e y )  =  1  ->  E. x  e.  RR*  ( B x e x )  =  A ) )
23223adant3 977 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( E. y  e.  RR*  ( B x e y )  =  1  ->  E. x  e.  RR*  ( B x e x )  =  A ) )
245, 23mpd 15 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. x  e.  RR*  ( B x e x )  =  A )
25 eqtr3 2408 . . . . . . 7  |-  ( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  ( B x e x )  =  ( B x e y ) )
26 simp1 957 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  x  e.  RR* )
27 simp2 958 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
y  e.  RR* )
28 simp3l 985 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  e.  RR )
29 simp3r 986 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  =/=  0 )
3026, 27, 28, 29xmulcand 24007 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( ( B x e x )  =  ( B x e y )  <->  x  =  y ) )
3125, 30syl5ib 211 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y )
)
32313expa 1153 . . . . 5  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( (
( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y ) )
3332expcom 425 . . . 4  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( (
( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y ) ) )
34333adant1 975 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y )
) )
3534ralrimivv 2742 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  A. x  e.  RR*  A. y  e. 
RR*  ( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y ) )
36 oveq2 6030 . . . 4  |-  ( x  =  y  ->  ( B x e x )  =  ( B x e y ) )
3736eqeq1d 2397 . . 3  |-  ( x  =  y  ->  (
( B x e x )  =  A  <-> 
( B x e y )  =  A ) )
3837reu4 3073 . 2  |-  ( E! x  e.  RR*  ( B x e x )  =  A  <->  ( E. x  e.  RR*  ( B x e x )  =  A  /\  A. x  e.  RR*  A. y  e.  RR*  ( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y ) ) )
3924, 35, 38sylanbrc 646 1  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B x e x )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   E!wreu 2653    C_ wss 3265  (class class class)co 6022   RRcr 8924   0cc0 8925   1c1 8926   RR*cxr 9054   x ecxmu 10643
This theorem is referenced by:  xdivcld  24009  xdivmul  24011  rexdiv  24012  xrmulc1cn  24122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-xneg 10644  df-xmul 10646
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