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Theorem receuap 8192
Description: Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
Assertion
Ref Expression
receuap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
Distinct variable groups:    x, A    x, B

Proof of Theorem receuap
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 recexap 8176 . . . 4  |-  ( ( B  e.  CC  /\  B #  0 )  ->  E. y  e.  CC  ( B  x.  y )  =  1 )
213adant1 962 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E. y  e.  CC  ( B  x.  y )  =  1 )
3 simprl 499 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
y  e.  CC )
4 simpll 497 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  A  e.  CC )
53, 4mulcld 7562 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( y  x.  A
)  e.  CC )
6 oveq1 5673 . . . . . . . 8  |-  ( ( B  x.  y )  =  1  ->  (
( B  x.  y
)  x.  A )  =  ( 1  x.  A ) )
76ad2antll 476 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( 1  x.  A ) )
8 simplr 498 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  B  e.  CC )
98, 3, 4mulassd 7565 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( B  x.  ( y  x.  A ) ) )
104mulid2d 7560 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
117, 9, 103eqtr3d 2129 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( B  x.  (
y  x.  A ) )  =  A )
12 oveq2 5674 . . . . . . . 8  |-  ( x  =  ( y  x.  A )  ->  ( B  x.  x )  =  ( B  x.  ( y  x.  A
) ) )
1312eqeq1d 2097 . . . . . . 7  |-  ( x  =  ( y  x.  A )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( y  x.  A
) )  =  A ) )
1413rspcev 2723 . . . . . 6  |-  ( ( ( y  x.  A
)  e.  CC  /\  ( B  x.  (
y  x.  A ) )  =  A )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
155, 11, 14syl2anc 404 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
1615rexlimdvaa 2491 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( B  x.  y )  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
17163adant3 964 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( E. y  e.  CC  ( B  x.  y
)  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
182, 17mpd 13 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  A )
19 eqtr3 2108 . . . . . . 7  |-  ( ( ( B  x.  x
)  =  A  /\  ( B  x.  y
)  =  A )  ->  ( B  x.  x )  =  ( B  x.  y ) )
20 mulcanap 8188 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  -> 
( ( B  x.  x )  =  ( B  x.  y )  <-> 
x  =  y ) )
2119, 20syl5ib 153 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
22213expa 1144 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  (
( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
2322expcom 115 . . . 4  |-  ( ( B  e.  CC  /\  B #  0 )  ->  (
( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
24233adant1 962 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
2524ralrimivv 2455 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A. x  e.  CC  A. y  e.  CC  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
26 oveq2 5674 . . . 4  |-  ( x  =  y  ->  ( B  x.  x )  =  ( B  x.  y ) )
2726eqeq1d 2097 . . 3  |-  ( x  =  y  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  y )  =  A ) )
2827reu4 2810 . 2  |-  ( E! x  e.  CC  ( B  x.  x )  =  A  <->  ( E. x  e.  CC  ( B  x.  x )  =  A  /\  A. x  e.  CC  A. y  e.  CC  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
2918, 25, 28sylanbrc 409 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 925    = wceq 1290    e. wcel 1439   A.wral 2360   E.wrex 2361   E!wreu 2362   class class class wbr 3851  (class class class)co 5666   CCcc 7402   0cc0 7404   1c1 7405    x. cmul 7409   # cap 8112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113
This theorem is referenced by:  divvalap  8195  divmulap  8196  divclap  8199
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