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Theorem negeu 8975
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
StepHypRef Expression
1 cnegex 8926 . . 3  |-  ( A  e.  CC  ->  E. y  e.  CC  ( A  +  y )  =  0 )
21adantr 453 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. y  e.  CC  ( A  +  y
)  =  0 )
3 simpl 445 . . . . . 6  |-  ( ( y  e.  CC  /\  ( A  +  y
)  =  0 )  ->  y  e.  CC )
4 simpr 449 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
5 ax-addcl 8730 . . . . . 6  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  +  B
)  e.  CC )
63, 4, 5syl2anr 466 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  -> 
( y  +  B
)  e.  CC )
7 simplrr 740 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  ( A  +  y )  =  0 )
87oveq1d 5772 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( 0  +  B ) )
9 simplll 737 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  A  e.  CC )
10 simplrl 739 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  y  e.  CC )
11 simpllr 738 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  e.  CC )
129, 10, 11addassd 8790 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( A  +  ( y  +  B
) ) )
1311addid2d 8946 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
0  +  B )  =  B )
148, 12, 133eqtr3rd 2297 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  =  ( A  +  ( y  +  B
) ) )
1514eqeq2d 2267 . . . . . . 7  |-  ( ( ( ( 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 449 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  x  e.  CC )
1710, 11addcld 8787 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
y  +  B )  e.  CC )
189, 16, 17addcand 8948 . . . . . . 7  |-  ( ( ( ( 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 246 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  B  <->  x  =  ( y  +  B
) ) )
2019ralrimiva 2597 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  ->  A. x  e.  CC  ( ( A  +  x )  =  B  <-> 
x  =  ( y  +  B ) ) )
21 reu6i 2909 . . . . 5  |-  ( ( ( y  +  B
)  e.  CC  /\  A. x  e.  CC  (
( A  +  x
)  =  B  <->  x  =  ( y  +  B
) ) )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
226, 20, 21syl2anc 645 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
2322expr 601 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  ->  E! x  e.  CC  ( A  +  x )  =  B ) )
2423rexlimdva 2638 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( A  +  y )  =  0  ->  E! x  e.  CC  ( A  +  x )  =  B ) )
252, 24mpd 16 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   E!wreu 2518  (class class class)co 5757   CCcc 8668   0cc0 8670    + caddc 8673
This theorem is referenced by:  subcl  8984  subadd  8987  addinv  20944
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-ltxr 8805
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