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Theorem negeu 9256
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 9207 . . 3  |-  ( A  e.  CC  ->  E. y  e.  CC  ( A  +  y )  =  0 )
21adantr 452 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. y  e.  CC  ( A  +  y
)  =  0 )
3 simpl 444 . . . 4  |-  ( ( y  e.  CC  /\  ( A  +  y
)  =  0 )  ->  y  e.  CC )
4 simpr 448 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
5 addcl 9032 . . . 4  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  +  B
)  e.  CC )
63, 4, 5syl2anr 465 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  -> 
( y  +  B
)  e.  CC )
7 simplrr 738 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  ( A  +  y )  =  0 )
87oveq1d 6059 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( 0  +  B ) )
9 simplll 735 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  A  e.  CC )
10 simplrl 737 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  y  e.  CC )
11 simpllr 736 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  e.  CC )
129, 10, 11addassd 9070 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  y )  +  B )  =  ( A  +  ( y  +  B
) ) )
1311addid2d 9227 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
0  +  B )  =  B )
148, 12, 133eqtr3rd 2449 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  B  =  ( A  +  ( y  +  B
) ) )
1514eqeq2d 2419 . . . . 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 448 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  x  e.  CC )
1710, 11addcld 9067 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
y  +  B )  e.  CC )
189, 16, 17addcand 9229 . . . . 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 245 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
y  e.  CC  /\  ( A  +  y
)  =  0 ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  B  <->  x  =  ( y  +  B
) ) )
2019ralrimiva 2753 . . 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 3089 . . 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 643 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( A  +  y )  =  0 ) )  ->  E! x  e.  CC  ( A  +  x
)  =  B )
232, 22rexlimddv 2798 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   E!wreu 2672  (class class class)co 6044   CCcc 8948   0cc0 8950    + caddc 8953
This theorem is referenced by:  subcl  9265  subadd  9268  addinv  21897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-ltxr 9085
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