ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnpcan Unicode version

Theorem nnpcan 7985
Description: Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
Assertion
Ref Expression
nnpcan  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  +  B )  =  ( A  -  C ) )

Proof of Theorem nnpcan
StepHypRef Expression
1 subcl 7961 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addsub 7973 . . . 4  |-  ( ( ( A  -  B
)  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( ( ( A  -  B )  -  C )  +  B ) )
43eqcomd 2145 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  +  B )  =  ( ( ( A  -  B )  +  B )  -  C ) )
52, 4syld3an1 1262 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  +  B )  =  ( ( ( A  -  B )  +  B )  -  C ) )
6 npcan 7971 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  B
)  =  A )
763adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  +  B )  =  A )
87oveq1d 5789 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  B
)  -  C )  =  ( A  -  C ) )
95, 8eqtrd 2172 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  C
)  +  B )  =  ( A  -  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7618    + caddc 7623    - cmin 7933
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator