Theorem negdii 7827

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

Step	Hyp	Ref	Expression
1		negidi.1	. . . . 5 |- A e. CC
2		pncan3i.2	. . . . 5 |- B e. CC
3	1, 2	addcli 7553	. . . 4 |- ( A + B ) e. CC
4	3	negidi 7812	. . 3 |- ( ( A + B ) + -u ( A + B ) ) = 0
5	1	negidi 7812	. . . . 5 |- ( A + -u A ) = 0
6	2	negidi 7812	. . . . 5 |- ( B + -u B ) = 0
7	5, 6	oveq12i 5678	. . . 4 |- ( ( A + -u A ) + ( B + -u B ) ) = ( 0 + 0 )
8		00id 7684	. . . 4 |- ( 0 + 0 ) = 0
9	7, 8	eqtri 2109	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = 0
10	1	negcli 7811	. . . 4 |- -u A e. CC
11	2	negcli 7811	. . . 4 |- -u B e. CC
12	1, 10, 2, 11	add4i 7708	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = ( ( A + B ) + ( -u A + -u B ) )
13	4, 9, 12	3eqtr2i 2115	. 2 |- ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) )
14	3	negcli 7811	. . 3 |- -u ( A + B ) e. CC
15	10, 11	addcli 7553	. . 3 |- ( -u A + -u B ) e. CC
16	3, 14, 15	addcani 7725	. 2 |- ( ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) ) <-> -u ( A + B ) = ( -u A + -u B ) )
17	13, 16	mpbi 144	1 |- -u ( A + B ) = ( -u A + -u B )

Syntax hints:   = wceq 1290   e. wcel 1439  (class class class)co 5666   CCcc 7409   0cc0 7411   + caddc 7414   -ucneg 7715

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366  ax-resscn 7498  ax-1cn 7499  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7716  df-neg 7717

This theorem is referenced by:  negsubdii  7828