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Theorem xreceu 24121
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 9085 . . . 4  |-  RR  C_  RR*
2 xrecex 24119 . . . . 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 3368 . . . 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 10837 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  -> 
( y x e A )  e.  RR* )
9 oveq1 6047 . . . . . . . 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 9090 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( y  e. 
RR*  /\  ( B x e y )  =  1 ) )  ->  B  e.  RR* )
13 xmulass 10822 . . . . . . . 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 10815 . . . . . . . 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 2444 . . . . . 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 6048 . . . . . . . 8  |-  ( x  =  ( y x e A )  -> 
( B x e x )  =  ( B x e ( y x e A ) ) )
1918eqeq1d 2412 . . . . . . 7  |-  ( x  =  ( y x e A )  -> 
( ( B x e x )  =  A  <->  ( B x e ( y x e A ) )  =  A ) )
2019rspcev 3012 . . . . . 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 2791 . . . 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 2423 . . . . . . 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 24120 . . . . . . 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 2757 . 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 6048 . . . 4  |-  ( x  =  y  ->  ( B x e x )  =  ( B x e y ) )
3736eqeq1d 2412 . . 3  |-  ( x  =  y  ->  (
( B x e x )  =  A  <-> 
( B x e y )  =  A ) )
3837reu4 3088 . 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 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   E!wreu 2668    C_ wss 3280  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947   RR*cxr 9075   x ecxmu 10665
This theorem is referenced by:  xdivcld  24122  xdivmul  24124  rexdiv  24125  xrmulc1cn  24269
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-xneg 10666  df-xmul 10668
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