MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addcan2 Unicode version

Theorem addcan2 9013
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 9009 . . 3  |-  ( C  e.  CC  ->  E. x  e.  CC  ( C  +  x )  =  0 )
213ad2ant3 978 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( C  +  x )  =  0 )
3 oveq1 5881 . . . . . 6  |-  ( ( A  +  C )  =  ( B  +  C )  ->  (
( A  +  C
)  +  x )  =  ( ( B  +  C )  +  x ) )
4 simpl1 958 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
5 simpl3 960 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
6 simprl 732 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
74, 5, 6addassd 8873 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  ( A  +  ( C  +  x ) ) )
8 simprr 733 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  x )  =  0 )
98oveq2d 5890 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  ( C  +  x ) )  =  ( A  +  0 ) )
10 addid1 9008 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
114, 10syl 15 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  0 )  =  A )
127, 9, 113eqtrd 2332 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  A )
13 simpl2 959 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
1413, 5, 6addassd 8873 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  ( B  +  ( C  +  x ) ) )
158oveq2d 5890 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  ( C  +  x ) )  =  ( B  +  0 ) )
16 addid1 9008 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B  +  0 )  =  B )
1713, 16syl 15 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  0 )  =  B )
1814, 15, 173eqtrd 2332 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  B )
1912, 18eqeq12d 2310 . . . . . 6  |-  ( ( ( 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 210 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  =  ( B  +  C )  ->  A  =  B ) )
21 oveq1 5881 . . . . 5  |-  ( A  =  B  ->  ( A  +  C )  =  ( B  +  C ) )
2220, 21impbid1 194 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
2322expr 598 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  x  e.  CC )  ->  ( ( C  +  x )  =  0  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) ) )
2423rexlimdva 2680 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. x  e.  CC  ( C  +  x
)  =  0  -> 
( ( A  +  C )  =  ( B  +  C )  <-> 
A  =  B ) ) )
252, 24mpd 14 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756
This theorem is referenced by:  addcom  9014  addcan2i  9022  addcomd  9030  addcan2d  9032  muleqadd  9428  subfacp1lem6  23731  axlowdimlem14  24655  trnij  25718  fargshiftf1  28382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888
  Copyright terms: Public domain W3C validator