Theorem negdii 8600

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 8320	. . . 4 |- ( A + B ) e. CC
4	3	negidi 8585	. . 3 |- ( ( A + B ) + -u ( A + B ) ) = 0
5	1	negidi 8585	. . . . 5 |- ( A + -u A ) = 0
6	2	negidi 8585	. . . . 5 |- ( B + -u B ) = 0
7	5, 6	oveq12i 6087	. . . 4 |- ( ( A + -u A ) + ( B + -u B ) ) = ( 0 + 0 )
8		00id 8457	. . . 4 |- ( 0 + 0 ) = 0
9	7, 8	eqtri 2259	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = 0
10	1	negcli 8584	. . . 4 |- -u A e. CC
11	2	negcli 8584	. . . 4 |- -u B e. CC
12	1, 10, 2, 11	add4i 8481	. . 3 |- ( ( A + -u A ) + ( B + -u B ) ) = ( ( A + B ) + ( -u A + -u B ) )
13	4, 9, 12	3eqtr2i 2265	. 2 |- ( ( A + B ) + -u ( A + B ) ) = ( ( A + B ) + ( -u A + -u B ) )
14	3	negcli 8584	. . 3 |- -u ( A + B ) e. CC
15	10, 11	addcli 8320	. . 3 |- ( -u A + -u B ) e. CC
16	3, 14, 15	addcani 8498	. 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 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   0cc0 8169   + caddc 8172   -ucneg 8488

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-neg 8490

This theorem is referenced by:  negsubdii  8601