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Theorem addcan 8127
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 8126 . . 3  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
213ad2ant1 1018 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( x  +  A )  =  0 )
3 oveq2 5877 . . . 4  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
4 simprr 531 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  A )  =  0 )
54oveq1d 5884 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( 0  +  B
) )
6 simprl 529 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  x  e.  CC )
7 simpl1 1000 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  A  e.  CC )
8 simpl2 1001 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  B  e.  CC )
96, 7, 8addassd 7970 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( x  +  ( A  +  B ) ) )
10 addid2 8086 . . . . . . 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 2218 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  B ) )  =  B )
134oveq1d 5884 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( 0  +  C
) )
14 simpl3 1002 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  C  e.  CC )
156, 7, 14addassd 7970 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( x  +  ( A  +  C ) ) )
16 addid2 8086 . . . . . . 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 2218 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  C ) )  =  C )
1912, 18eqeq12d 2192 . . . 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, 19imbitrid 154 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) )
21 oveq2 5877 . . 3  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
2220, 21impbid1 142 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
232, 22rexlimddv 2599 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456  (class class class)co 5869   CCcc 7800   0cc0 7802    + caddc 7805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872
This theorem is referenced by:  addcani  8129  addcand  8131  subcan  8202
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