Theorem negdii 8305

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 8025	. . . 4 |- ( A + B ) e. CC
4	3	negidi 8290	. . 3 |- ( ( A + B ) + -u ( A + B ) ) = 0
5	1	negidi 8290	. . . . 5 |- ( A + -u A ) = 0
6	2	negidi 8290	. . . . 5 |- ( B + -u B ) = 0
7	5, 6	oveq12i 5931	. . . 4 |- ( ( A + -u A ) + ( B + -u B ) ) = ( 0 + 0 )
8		00id 8162	. . . 4 |- ( 0 + 0 ) = 0
9	7, 8	eqtri 2214	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = 0
10	1	negcli 8289	. . . 4 |- -u A e. CC
11	2	negcli 8289	. . . 4 |- -u B e. CC
12	1, 10, 2, 11	add4i 8186	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = ( ( A + B ) + ( -u A + -u B ) )
13	4, 9, 12	3eqtr2i 2220	. 2 |- ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) )
14	3	negcli 8289	. . 3 |- -u ( A + B ) e. CC
15	10, 11	addcli 8025	. . 3 |- ( -u A + -u B ) e. CC
16	3, 14, 15	addcani 8203	. 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 1364   e. wcel 2164  (class class class)co 5919   CCcc 7872   0cc0 7874   + caddc 7877   -ucneg 8193

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194  df-neg 8195

This theorem is referenced by:  negsubdii  8306