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Theorem npncan 8178
Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
Assertion
Ref Expression
npncan  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  +  ( B  -  C ) )  =  ( A  -  C ) )

Proof of Theorem npncan
StepHypRef Expression
1 subcl 8156 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addsubass 8167 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( ( A  -  B )  +  ( B  -  C
) ) )
42, 3syld3an1 1284 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( ( A  -  B )  +  ( B  -  C
) ) )
5 npcan 8166 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  B
)  =  A )
65oveq1d 5890 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B )  +  B )  -  C
)  =  ( A  -  C ) )
763adant3 1017 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( A  -  C ) )
84, 7eqtr3d 2212 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  +  ( B  -  C ) )  =  ( A  -  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148  (class class class)co 5875   CCcc 7809    + caddc 7814    - cmin 8128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-sub 8130
This theorem is referenced by:  npncan2  8184  npncand  8292  bcn2  10744  abs3dif  11114
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