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Theorem xreceu 23121
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 8892 . . . 4  |-  RR  C_  RR*
2 xrecex 23119 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B x e y )  =  1 )
323adant1 973 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR  ( B x e y )  =  1 )
4 ssrexv 3251 . . . 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 59 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. y  e.  RR*  ( B x e y )  =  1 )
6 simprl 732 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
y  e.  RR* )
7 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  A  e.  RR* )
86, 7xmulcld 10638 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( y x e A )  e.  RR* )
9 oveq1 5881 . . . . . . . . 9  |-  ( ( B x e y )  =  1  -> 
( ( B x e y ) x e A )  =  ( 1 x e A ) )
109ad2antll 709 . . . . . . . 8  |-  ( ( ( 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 731 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  B  e.  RR )
1211rexrd 8897 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  B  e.  RR* )
13 xmulass 10623 . . . . . . . . 9  |-  ( ( 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 1182 . . . . . . . 8  |-  ( ( ( 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 10616 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( 1 x e A )  =  A )
167, 15syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( 1 x e A )  =  A )
1710, 14, 163eqtr3d 2336 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( B x e ( y x e A ) )  =  A )
18 oveq2 5882 . . . . . . . . 9  |-  ( x  =  ( y x e A )  -> 
( B x e x )  =  ( B x e ( y x e A ) ) )
1918eqeq1d 2304 . . . . . . . 8  |-  ( x  =  ( y x e A )  -> 
( ( B x e x )  =  A  <->  ( B x e ( y x e A ) )  =  A ) )
2019rspcev 2897 . . . . . . 7  |-  ( ( ( 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 642 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  E. x  e.  RR*  ( B x e x )  =  A )
2221exp32 588 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
y  e.  RR*  ->  ( ( B x e y )  =  1  ->  E. x  e.  RR*  ( B x e x )  =  A ) ) )
2322rexlimdv 2679 . . . 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 ) )
24233adant3 975 . . 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 ) )
255, 24mpd 14 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. x  e.  RR*  ( B x e x )  =  A )
26 eqtr3 2315 . . . . . . 7  |-  ( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  ( B x e x )  =  ( B x e y ) )
27 simp1 955 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  x  e.  RR* )
28 simp2 956 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
y  e.  RR* )
29 simp3l 983 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  e.  RR )
30 simp3r 984 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  B  =/=  0 )
3127, 28, 29, 30xmulcand 23120 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( ( B x e x )  =  ( B x e y )  <->  x  =  y ) )
3226, 31syl5ib 210 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( ( ( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y )
)
33323expa 1151 . . . . 5  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( (
( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y ) )
3433expcom 424 . . . 4  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( (
( B x e x )  =  A  /\  ( B x e y )  =  A )  ->  x  =  y ) ) )
35343adant1 973 . . 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 )
) )
3635ralrimivv 2647 . 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 ) )
37 oveq2 5882 . . . 4  |-  ( x  =  y  ->  ( B x e x )  =  ( B x e y ) )
3837eqeq1d 2304 . . 3  |-  ( x  =  y  ->  (
( B x e x )  =  A  <-> 
( B x e y )  =  A ) )
3938reu4 2972 . 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 ) ) )
4025, 36, 39sylanbrc 645 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558    C_ wss 3165  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754   RR*cxr 8882   x ecxmu 10467
This theorem is referenced by:  xdivcld  23122  xdivmul  23124  rexdiv  23125  xrmulc1cn  23318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-xneg 10468  df-xmul 10470
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