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Theorem addcan 7906
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 7905 . . 3  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
213ad2ant1 985 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( x  +  A )  =  0 )
3 oveq2 5748 . . . 4  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
4 simprr 504 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  A )  =  0 )
54oveq1d 5755 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( 0  +  B
) )
6 simprl 503 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  x  e.  CC )
7 simpl1 967 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  A  e.  CC )
8 simpl2 968 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  B  e.  CC )
96, 7, 8addassd 7752 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( x  +  ( A  +  B ) ) )
10 addid2 7865 . . . . . . 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 2156 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  B ) )  =  B )
134oveq1d 5755 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( 0  +  C
) )
14 simpl3 969 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  C  e.  CC )
156, 7, 14addassd 7752 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( x  +  ( A  +  C ) ) )
16 addid2 7865 . . . . . . 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 2156 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  C ) )  =  C )
1912, 18eqeq12d 2130 . . . 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 153 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) )
21 oveq2 5748 . . 3  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
2220, 21impbid1 141 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
232, 22rexlimddv 2529 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392  (class class class)co 5740   CCcc 7582   0cc0 7584    + caddc 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-resscn 7676  ax-1cn 7677  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743
This theorem is referenced by:  addcani  7908  addcand  7910  subcan  7981
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