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Theorem addcan2 8997
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 8993 . . 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 5865 . . . . . 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 8857 . . . . . . . 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 5874 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  ( C  +  x ) )  =  ( A  +  0 ) )
10 addid1 8992 . . . . . . . . 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 2319 . . . . . . 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 8857 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  ( B  +  ( C  +  x ) ) )
158oveq2d 5874 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  ( C  +  x ) )  =  ( B  +  0 ) )
16 addid1 8992 . . . . . . . . 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 2319 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  B )
1912, 18eqeq12d 2297 . . . . . 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 5865 . . . . 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 2667 . 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 1623    e. wcel 1684   E.wrex 2544  (class class class)co 5858   CCcc 8735   0cc0 8737    + caddc 8740
This theorem is referenced by:  addcom  8998  addcan2i  9006  addcomd  9014  addcan2d  9016  muleqadd  9412  subfacp1lem6  23716  axlowdimlem14  24583  trnij  25615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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