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Theorem xreceu 29630
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 10083 . . . 4 |- RR C_ RR*
2 xrecex 29628 . . . . 5 |- ( ( B e. RR /\ B =/= 0 ) -> E. y e. RR ( B xe y ) = 1 )
323adant1 1079 . . . 4 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> E. y e. RR ( B xe y ) = 1 )
4 ssrexv 3667 . . . 4 |- ( RR C_ RR* -> ( E. y e. RR ( B xe y ) = 1 -> E. y e. RR* ( B xe y ) = 1 ) )
51, 3, 4mpsyl 68 . . 3 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> E. y e. RR* ( B xe y ) = 1 )
6 simprl 794 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> y e. RR* )
7 simpll 790 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> A e. RR* )
86, 7xmulcld 12132 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> ( y xe A ) e. RR* )
9 oveq1 6657 . . . . . . . 8 |- ( ( B xe y ) = 1 -> ( ( B xe y ) xe A ) = ( 1 xe A ) )
109ad2antll 765 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> ( ( B xe y ) xe A ) = ( 1 xe A ) )
11 simplr 792 . . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> B e. RR )
1211rexrd 10089 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> B e. RR* )
13 xmulass 12117 . . . . . . . 8 |- ( ( B e. RR* /\ y e. RR* /\ A e. RR* ) -> ( ( B xe y ) xe A ) = ( B xe ( y xe A ) ) )
1412, 6, 7, 13syl3anc 1326 . . . . . . 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 12110 . . . . . . . 8 |- ( A e. RR* -> ( 1 xe A ) = A )
167, 15syl 17 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> ( 1 xe A ) = A )
1710, 14, 163eqtr3d 2664 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> ( B xe ( y xe A ) ) = A )
18 oveq2 6658 . . . . . . . 8 |- ( x = ( y xe A ) -> ( B xe x ) = ( B xe ( y xe A ) ) )
1918eqeq1d 2624 . . . . . . 7 |- ( x = ( y xe A ) -> ( ( B xe x ) = A <-> ( B xe ( y xe A ) ) = A ) )
2019rspcev 3309 . . . . . 6 |- ( ( ( y xe A ) e. RR* /\ ( B xe ( y xe A ) ) = A ) -> E. x e. RR* ( B xe x ) = A )
218, 17, 20syl2anc 693 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( y e. RR* /\ ( B xe y ) = 1 ) ) -> E. x e. RR* ( B xe x ) = A )
2221rexlimdvaa 3032 . . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( E. y e. RR* ( B xe y ) = 1 -> E. x e. RR* ( B xe x ) = A ) )
23223adant3 1081 . . 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 2643 . . . . . . 7 |- ( ( ( B xe x ) = A /\ ( B xe y ) = A ) -> ( B xe x ) = ( B xe y ) )
26 simp1 1061 . . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> x e. RR* )
27 simp2 1062 . . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> y e. RR* )
28 simp3l 1089 . . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> B e. RR )
29 simp3r 1090 . . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> B =/= 0 )
3026, 27, 28, 29xmulcand 29629 . . . . . . 7 |- ( ( x e. RR* /\ y e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( B xe x ) = ( B xe y ) <-> x = y ) )
3125, 30syl5ib 234 . . . . . 6 |- ( ( x e. RR* /\ y e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( ( B xe x ) = A /\ ( B xe y ) = A ) -> x = y ) )
32313expa 1265 . . . . 5 |- ( ( ( x e. RR* /\ y e. RR* ) /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( ( B xe x ) = A /\ ( B xe y ) = A ) -> x = y ) )
3332expcom 451 . . . 4 |- ( ( B e. RR /\ B =/= 0 ) -> ( ( x e. RR* /\ y e. RR* ) -> ( ( ( B xe x ) = A /\ ( B xe y ) = A ) -> x = y ) ) )
34333adant1 1079 . . 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 2970 . 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 6658 . . . 4 |- ( x = y -> ( B xe x ) = ( B xe y ) )
3736eqeq1d 2624 . . 3 |- ( x = y -> ( ( B xe x ) = A <-> ( B xe y ) = A ) )
3837reu4 3400 . 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 698 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   C_ wss 3574  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   xecxmu 11945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946  df-xmul 11948
This theorem is referenced by:  xdivcld  29631  xdivmul  29633  rexdiv  29634  xrmulc1cn  29976
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