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Theorem addcan2 9765
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 9761 . . 3  |-  ( C  e.  CC  ->  E. x  e.  CC  ( C  +  x )  =  0 )
213ad2ant3 1019 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( C  +  x )  =  0 )
3 oveq1 6292 . . . 4  |-  ( ( A  +  C )  =  ( B  +  C )  ->  (
( A  +  C
)  +  x )  =  ( ( B  +  C )  +  x ) )
4 simpl1 999 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
5 simpl3 1001 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
6 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
74, 5, 6addassd 9619 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  ( A  +  ( C  +  x ) ) )
8 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  x )  =  0 )
98oveq2d 6301 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  ( C  +  x ) )  =  ( A  +  0 ) )
10 addid1 9760 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
114, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( A  +  0 )  =  A )
127, 9, 113eqtrd 2512 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  +  x )  =  A )
13 simpl2 1000 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
1413, 5, 6addassd 9619 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  ( B  +  ( C  +  x ) ) )
158oveq2d 6301 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  ( C  +  x ) )  =  ( B  +  0 ) )
16 addid1 9760 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  +  0 )  =  B )
1713, 16syl 16 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( B  +  0 )  =  B )
1814, 15, 173eqtrd 2512 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( B  +  C )  +  x )  =  B )
1912, 18eqeq12d 2489 . . . 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 219 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  =  ( B  +  C )  ->  A  =  B ) )
21 oveq1 6292 . . 3  |-  ( A  =  B  ->  ( A  +  C )  =  ( B  +  C ) )
2220, 21impbid1 203 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( C  +  x
)  =  0 ) )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
232, 22rexlimddv 2959 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  =  ( B  +  C )  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815  (class class class)co 6285   CCcc 9491   0cc0 9493    + caddc 9496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-ltxr 9634
This theorem is referenced by:  addcom  9766  addcan2i  9774  addcomd  9782  addcan2d  9784  muleqadd  10194  axlowdimlem14  24031  fargshiftf1  24410  subfacp1lem6  28380
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