Theorem negdii 8462

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 8182	. . . 4 |- ( A + B ) e. CC
4	3	negidi 8447	. . 3 |- ( ( A + B ) + -u ( A + B ) ) = 0
5	1	negidi 8447	. . . . 5 |- ( A + -u A ) = 0
6	2	negidi 8447	. . . . 5 |- ( B + -u B ) = 0
7	5, 6	oveq12i 6029	. . . 4 |- ( ( A + -u A ) + ( B + -u B ) ) = ( 0 + 0 )
8		00id 8319	. . . 4 |- ( 0 + 0 ) = 0
9	7, 8	eqtri 2252	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = 0
10	1	negcli 8446	. . . 4 |- -u A e. CC
11	2	negcli 8446	. . . 4 |- -u B e. CC
12	1, 10, 2, 11	add4i 8343	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = ( ( A + B ) + ( -u A + -u B ) )
13	4, 9, 12	3eqtr2i 2258	. 2 |- ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) )
14	3	negcli 8446	. . 3 |- -u ( A + B ) e. CC
15	10, 11	addcli 8182	. . 3 |- ( -u A + -u B ) e. CC
16	3, 14, 15	addcani 8360	. 2 |- ( ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) ) <-> -u ( A + B ) = ( -u A + -u B ) )
17	13, 16	mpbi 145	1 |- -u ( A + B ) = ( -u A + -u B )

Syntax hints:   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   0cc0 8031   + caddc 8034   -ucneg 8350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351  df-neg 8352

This theorem is referenced by:  negsubdii  8463