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Theorem negeu 9038
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 8989 . . 3  |-  ( A  e.  CC  ->  E. y  e.  CC  ( A  +  y )  =  0 )
21adantr 451 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. y  e.  CC  ( A  +  y
)  =  0 )
3 simpl 443 . . . . . 6  |-  ( ( y  e.  CC  /\  ( A  +  y
)  =  0 )  ->  y  e.  CC )
4 simpr 447 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
5 addcl 8815 . . . . . 6  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  +  B
)  e.  CC )
63, 4, 5syl2anr 464 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  -> 
( y  +  B
)  e.  CC )
7 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  ( A  +  y )  =  0 )
87oveq1d 5835 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( 0  +  B ) )
9 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  A  e.  CC )
10 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  y  e.  CC )
11 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  e.  CC )
129, 10, 11addassd 8853 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( A  +  ( y  +  B
) ) )
1311addid2d 9009 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
0  +  B )  =  B )
148, 12, 133eqtr3rd 2325 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  =  ( A  +  ( y  +  B
) ) )
1514eqeq2d 2295 . . . . . . 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 447 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  x  e.  CC )
1710, 11addcld 8850 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
y  +  B )  e.  CC )
189, 16, 17addcand 9011 . . . . . . 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 244 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  B  <->  x  =  ( y  +  B
) ) )
2019ralrimiva 2627 . . . . 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 2957 . . . . 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 642 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
2322expr 598 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  ->  E! x  e.  CC  ( A  +  x )  =  B ) )
2423rexlimdva 2668 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( A  +  y )  =  0  ->  E! x  e.  CC  ( A  +  x )  =  B ) )
252, 24mpd 14 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   E.wrex 2545   E!wreu 2546  (class class class)co 5820   CCcc 8731   0cc0 8733    + caddc 8736
This theorem is referenced by:  subcl  9047  subadd  9050  addinv  21013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868
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