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Theorem addcan 9565
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addcan  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )

Proof of Theorem addcan
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnegex2 9563 . . 3  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
213ad2ant1 1009 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( x  +  A )  =  0 )
3 oveq2 6111 . . . 4  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
4 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  A )  =  0 )
54oveq1d 6118 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( 0  +  B
) )
6 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  x  e.  CC )
7 simpl1 991 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  A  e.  CC )
8 simpl2 992 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  B  e.  CC )
96, 7, 8addassd 9420 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( x  +  ( A  +  B ) ) )
10 addid2 9564 . . . . . . 7  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
118, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( 0  +  B )  =  B )
125, 9, 113eqtr3d 2483 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  B ) )  =  B )
134oveq1d 6118 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( 0  +  C
) )
14 simpl3 993 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  C  e.  CC )
156, 7, 14addassd 9420 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( x  +  ( A  +  C ) ) )
16 addid2 9564 . . . . . . 7  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
1714, 16syl 16 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( 0  +  C )  =  C )
1813, 15, 173eqtr3d 2483 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  C ) )  =  C )
1912, 18eqeq12d 2457 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) )  <->  B  =  C ) )
203, 19syl5ib 219 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) )
21 oveq2 6111 . . 3  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
2220, 21impbid1 203 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
232, 22rexlimddv 2857 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2728  (class class class)co 6103   CCcc 9292   0cc0 9294    + caddc 9297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-ltxr 9435
This theorem is referenced by:  addcom  9567  addcani  9574  addcomd  9583  addcand  9584  subcan  9676
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