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Theorem addcan 7254
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addcan  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )

Proof of Theorem addcan
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnegex2 7253 . . 3  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
213ad2ant1 936 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( x  +  A )  =  0 )
3 oveq2 5548 . . . 4  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
4 simprr 492 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  A )  =  0 )
54oveq1d 5555 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( 0  +  B
) )
6 simprl 491 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  x  e.  CC )
7 simpl1 918 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  A  e.  CC )
8 simpl2 919 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  B  e.  CC )
96, 7, 8addassd 7107 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( x  +  ( A  +  B ) ) )
10 addid2 7213 . . . . . . 7  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
118, 10syl 14 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( 0  +  B )  =  B )
125, 9, 113eqtr3d 2096 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  B ) )  =  B )
134oveq1d 5555 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( 0  +  C
) )
14 simpl3 920 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  C  e.  CC )
156, 7, 14addassd 7107 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( x  +  ( A  +  C ) ) )
16 addid2 7213 . . . . . . 7  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
1714, 16syl 14 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( 0  +  C )  =  C )
1813, 15, 173eqtr3d 2096 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  C ) )  =  C )
1912, 18eqeq12d 2070 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) )  <->  B  =  C ) )
203, 19syl5ib 147 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) )
21 oveq2 5548 . . 3  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
2220, 21impbid1 134 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
232, 22rexlimddv 2454 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324  (class class class)co 5540   CCcc 6945   0cc0 6947    + caddc 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  addcani  7256  addcand  7258  subcan  7329
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