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Theorem addcan2 8100
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 8097 . . 3  |-  ( C  e.  CC  ->  E. x  e.  CC  ( C  +  x )  =  0 )
213ad2ant3 1015 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( C  +  x )  =  0 )
3 oveq1 5860 . . . 4  |-  ( ( A  +  C )  =  ( B  +  C )  ->  (
( A  +  C
)  +  x )  =  ( ( B  +  C )  +  x ) )
4 simpl1 995 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
5 simpl3 997 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
6 simprl 526 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
74, 5, 6addassd 7942 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  ( A  +  ( C  +  x ) ) )
8 simprr 527 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  x )  =  0 )
98oveq2d 5869 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  ( C  +  x ) )  =  ( A  +  0 ) )
10 addid1 8057 . . . . . . 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 2207 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  A )
13 simpl2 996 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
1413, 5, 6addassd 7942 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  ( B  +  ( C  +  x ) ) )
158oveq2d 5869 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  ( C  +  x ) )  =  ( B  +  0 ) )
16 addid1 8057 . . . . . . 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 2207 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  B )
1912, 18eqeq12d 2185 . . . 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 5860 . . 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 2592 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449  (class class class)co 5853   CCcc 7772   0cc0 7774    + caddc 7777
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by:  addcan2i  8102  addcan2d  8104  muleqadd  8586
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