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Theorem negeu 7671
Description: Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
negeu  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem negeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnegex 7658 . . 3  |-  ( A  e.  CC  ->  E. y  e.  CC  ( A  +  y )  =  0 )
21adantr 270 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. y  e.  CC  ( A  +  y
)  =  0 )
3 simpl 107 . . . 4  |-  ( ( y  e.  CC  /\  ( A  +  y
)  =  0 )  ->  y  e.  CC )
4 simpr 108 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
5 addcl 7465 . . . 4  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  +  B
)  e.  CC )
63, 4, 5syl2anr 284 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  -> 
( y  +  B
)  e.  CC )
7 simplrr 503 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  ( A  +  y )  =  0 )
87oveq1d 5667 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( 0  +  B ) )
9 simplll 500 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  A  e.  CC )
10 simplrl 502 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  y  e.  CC )
11 simpllr 501 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  e.  CC )
129, 10, 11addassd 7508 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( A  +  ( y  +  B
) ) )
1311addid2d 7630 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
0  +  B )  =  B )
148, 12, 133eqtr3rd 2129 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  =  ( A  +  ( y  +  B
) ) )
1514eqeq2d 2099 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  B  <->  ( A  +  x )  =  ( A  +  ( y  +  B ) ) ) )
16 simpr 108 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  x  e.  CC )
1710, 11addcld 7505 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
y  +  B )  e.  CC )
189, 16, 17addcand 7664 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  ( A  +  ( y  +  B ) )  <->  x  =  ( y  +  B
) ) )
1915, 18bitrd 186 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  B  <->  x  =  ( y  +  B
) ) )
2019ralrimiva 2446 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  ->  A. x  e.  CC  ( ( A  +  x )  =  B  <-> 
x  =  ( y  +  B ) ) )
21 reu6i 2806 . . 3  |-  ( ( ( y  +  B
)  e.  CC  /\  A. x  e.  CC  (
( A  +  x
)  =  B  <->  x  =  ( y  +  B
) ) )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
226, 20, 21syl2anc 403 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
232, 22rexlimddv 2493 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   E!wreu 2361  (class class class)co 5652   CCcc 7346   0cc0 7348    + caddc 7351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  subval  7672  subcl  7679  subadd  7683
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