Theorem negdii 8391

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 8111	. . . 4 |- ( A + B ) e. CC
4	3	negidi 8376	. . 3 |- ( ( A + B ) + -u ( A + B ) ) = 0
5	1	negidi 8376	. . . . 5 |- ( A + -u A ) = 0
6	2	negidi 8376	. . . . 5 |- ( B + -u B ) = 0
7	5, 6	oveq12i 5979	. . . 4 |- ( ( A + -u A ) + ( B + -u B ) ) = ( 0 + 0 )
8		00id 8248	. . . 4 |- ( 0 + 0 ) = 0
9	7, 8	eqtri 2228	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = 0
10	1	negcli 8375	. . . 4 |- -u A e. CC
11	2	negcli 8375	. . . 4 |- -u B e. CC
12	1, 10, 2, 11	add4i 8272	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = ( ( A + B ) + ( -u A + -u B ) )
13	4, 9, 12	3eqtr2i 2234	. 2 |- ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) )
14	3	negcli 8375	. . 3 |- -u ( A + B ) e. CC
15	10, 11	addcli 8111	. . 3 |- ( -u A + -u B ) e. CC
16	3, 14, 15	addcani 8289	. 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 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   0cc0 7960   + caddc 7963   -ucneg 8279

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-neg 8281

This theorem is referenced by:  negsubdii  8392