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Theorem addcan 8876
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
StepHypRef Expression
1 cnegex2 8874 . . 3  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
213ad2ant1 981 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  CC  ( x  +  A )  =  0 )
3 oveq2 5718 . . . . . 6  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
4 simprr 736 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  A )  =  0 )
54oveq1d 5725 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( 0  +  B
) )
6 simprl 735 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  x  e.  CC )
7 simpl1 963 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  A  e.  CC )
8 simpl2 964 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  B  e.  CC )
96, 7, 8addassd 8737 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  B )  =  ( x  +  ( A  +  B ) ) )
10 addid2 8875 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
118, 10syl 17 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( 0  +  B )  =  B )
125, 9, 113eqtr3d 2293 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  B ) )  =  B )
134oveq1d 5725 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( 0  +  C
) )
14 simpl3 965 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  C  e.  CC )
156, 7, 14addassd 8737 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( (
x  +  A )  +  C )  =  ( x  +  ( A  +  C ) ) )
16 addid2 8875 . . . . . . . . 9  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
1714, 16syl 17 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( 0  +  C )  =  C )
1813, 15, 173eqtr3d 2293 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( x  +  ( A  +  C ) )  =  C )
1912, 18eqeq12d 2267 . . . . . 6  |-  ( ( ( 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 212 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) )
21 oveq2 5718 . . . . 5  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
2220, 21impbid1 196 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( x  e.  CC  /\  ( x  +  A
)  =  0 ) )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
2322expr 601 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  x  e.  CC )  ->  ( ( x  +  A )  =  0  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) ) )
2423rexlimdva 2629 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. x  e.  CC  ( x  +  A
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  <-> 
B  =  C ) ) )
252, 24mpd 16 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2510  (class class class)co 5710   CCcc 8615   0cc0 8617    + caddc 8620
This theorem is referenced by:  addcom  8878  addcani  8885  addcomd  8894  addcand  8895  subcan  8982
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752
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