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Theorem addcan2 8056
Description: Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
addcan2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  =  ( B  +  C )  <->  A  =  B ) )

Proof of Theorem addcan2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnegex 8053 . . 3  |-  ( C  e.  CC  ->  E. x  e.  CC  ( C  +  x )  =  0 )
213ad2ant3 1005 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( C  +  x )  =  0 )
3 oveq1 5831 . . . 4  |-  ( ( A  +  C )  =  ( B  +  C )  ->  (
( A  +  C
)  +  x )  =  ( ( B  +  C )  +  x ) )
4 simpl1 985 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
5 simpl3 987 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
6 simprl 521 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
74, 5, 6addassd 7900 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  ( A  +  ( C  +  x ) ) )
8 simprr 522 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  x )  =  0 )
98oveq2d 5840 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  ( C  +  x ) )  =  ( A  +  0 ) )
10 addid1 8013 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
114, 10syl 14 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  0 )  =  A )
127, 9, 113eqtrd 2194 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  A )
13 simpl2 986 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
1413, 5, 6addassd 7900 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  ( B  +  ( C  +  x ) ) )
158oveq2d 5840 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  ( C  +  x ) )  =  ( B  +  0 ) )
16 addid1 8013 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  +  0 )  =  B )
1713, 16syl 14 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  0 )  =  B )
1814, 15, 173eqtrd 2194 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  B )
1912, 18eqeq12d 2172 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( (
( A  +  C
)  +  x )  =  ( ( B  +  C )  +  x )  <->  A  =  B ) )
203, 19syl5ib 153 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  =  ( B  +  C )  ->  A  =  B ) )
21 oveq1 5831 . . 3  |-  ( A  =  B  ->  ( A  +  C )  =  ( B  +  C ) )
2220, 21impbid1 141 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
232, 22rexlimddv 2579 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  =  ( B  +  C )  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   E.wrex 2436  (class class class)co 5824   CCcc 7730   0cc0 7732    + caddc 7735
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5135  df-fv 5178  df-ov 5827
This theorem is referenced by:  addcan2i  8058  addcan2d  8060  muleqadd  8542
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