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Theorem negeu 9044
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 8995 . . 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 8821 . . . . . 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 5875 . . . . . . . . 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 8859 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( A  +  ( y  +  B
) ) )
1311addid2d 9015 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
0  +  B )  =  B )
148, 12, 133eqtr3rd 2326 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  =  ( A  +  ( y  +  B
) ) )
1514eqeq2d 2296 . . . . . . 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 8856 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
y  +  B )  e.  CC )
189, 16, 17addcand 9017 . . . . . . 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 2628 . . . . 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 2958 . . . . 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 2669 . 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 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   E!wreu 2547  (class class class)co 5860   CCcc 8737   0cc0 8739    + caddc 8742
This theorem is referenced by:  subcl  9053  subadd  9056  addinv  21021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874
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