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

Theorem negdii 8058
Description: Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Hypotheses
Ref Expression
negidi.1  |-  A  e.  CC
pncan3i.2  |-  B  e.  CC
Assertion
Ref Expression
negdii  |-  -u ( A  +  B )  =  ( -u A  +  -u B )

Proof of Theorem negdii
StepHypRef Expression
1 negidi.1 . . . . 5  |-  A  e.  CC
2 pncan3i.2 . . . . 5  |-  B  e.  CC
31, 2addcli 7782 . . . 4  |-  ( A  +  B )  e.  CC
43negidi 8043 . . 3  |-  ( ( A  +  B )  +  -u ( A  +  B ) )  =  0
51negidi 8043 . . . . 5  |-  ( A  +  -u A )  =  0
62negidi 8043 . . . . 5  |-  ( B  +  -u B )  =  0
75, 6oveq12i 5786 . . . 4  |-  ( ( A  +  -u A
)  +  ( B  +  -u B ) )  =  ( 0  +  0 )
8 00id 7915 . . . 4  |-  ( 0  +  0 )  =  0
97, 8eqtri 2160 . . 3  |-  ( ( A  +  -u A
)  +  ( B  +  -u B ) )  =  0
101negcli 8042 . . . 4  |-  -u A  e.  CC
112negcli 8042 . . . 4  |-  -u B  e.  CC
121, 10, 2, 11add4i 7939 . . 3  |-  ( ( A  +  -u A
)  +  ( B  +  -u B ) )  =  ( ( A  +  B )  +  ( -u A  +  -u B ) )
134, 9, 123eqtr2i 2166 . 2  |-  ( ( A  +  B )  +  -u ( A  +  B ) )  =  ( ( A  +  B )  +  (
-u A  +  -u B ) )
143negcli 8042 . . 3  |-  -u ( A  +  B )  e.  CC
1510, 11addcli 7782 . . 3  |-  ( -u A  +  -u B )  e.  CC
163, 14, 15addcani 7956 . 2  |-  ( ( ( A  +  B
)  +  -u ( A  +  B )
)  =  ( ( A  +  B )  +  ( -u A  +  -u B ) )  <->  -u ( A  +  B
)  =  ( -u A  +  -u B ) )
1713, 16mpbi 144 1  |-  -u ( A  +  B )  =  ( -u A  +  -u B )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7630   0cc0 7632    + caddc 7635   -ucneg 7946
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 7724  ax-1cn 7725  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
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 7947  df-neg 7948
This theorem is referenced by:  negsubdii  8059
  Copyright terms: Public domain W3C validator