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Theorem npncan 8127
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 8105 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 1012 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addsubass 8116 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( ( A  -  B )  +  ( B  -  C
) ) )
42, 3syld3an1 1279 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( ( A  -  B )  +  ( B  -  C
) ) )
5 npcan 8115 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  B
)  =  A )
65oveq1d 5865 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B )  +  B )  -  C
)  =  ( A  -  C ) )
763adant3 1012 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( A  -  C ) )
84, 7eqtr3d 2205 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 103    /\ w3a 973    = wceq 1348    e. wcel 2141  (class class class)co 5850   CCcc 7759    + caddc 7764    - cmin 8077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-resscn 7853  ax-1cn 7854  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-sub 8079
This theorem is referenced by:  npncan2  8133  npncand  8241  bcn2  10685  abs3dif  11056
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