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Theorem negdii 8203
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 7924 . . . 4  |-  ( A  +  B )  e.  CC
43negidi 8188 . . 3  |-  ( ( A  +  B )  +  -u ( A  +  B ) )  =  0
51negidi 8188 . . . . 5  |-  ( A  +  -u A )  =  0
62negidi 8188 . . . . 5  |-  ( B  +  -u B )  =  0
75, 6oveq12i 5865 . . . 4  |-  ( ( A  +  -u A
)  +  ( B  +  -u B ) )  =  ( 0  +  0 )
8 00id 8060 . . . 4  |-  ( 0  +  0 )  =  0
97, 8eqtri 2191 . . 3  |-  ( ( A  +  -u A
)  +  ( B  +  -u B ) )  =  0
101negcli 8187 . . . 4  |-  -u A  e.  CC
112negcli 8187 . . . 4  |-  -u B  e.  CC
121, 10, 2, 11add4i 8084 . . 3  |-  ( ( A  +  -u A
)  +  ( B  +  -u B ) )  =  ( ( A  +  B )  +  ( -u A  +  -u B ) )
134, 9, 123eqtr2i 2197 . 2  |-  ( ( A  +  B )  +  -u ( A  +  B ) )  =  ( ( A  +  B )  +  (
-u A  +  -u B ) )
143negcli 8187 . . 3  |-  -u ( A  +  B )  e.  CC
1510, 11addcli 7924 . . 3  |-  ( -u A  +  -u B )  e.  CC
163, 14, 15addcani 8101 . 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 1348    e. wcel 2141  (class class class)co 5853   CCcc 7772   0cc0 7774    + caddc 7777   -ucneg 8091
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 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093
This theorem is referenced by:  negsubdii  8204
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