Theorem negsubdii 8558

Description: Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)

Hypotheses

Ref	Expression
negidi.1	|- A e. CC
pncan3i.2	|- B e. CC

Assertion

Ref	Expression
negsubdii	|- -u ( A - B ) = ( -u A + B )

Proof of Theorem negsubdii

Step	Hyp	Ref	Expression
1		negidi.1	. . 3 |- A e. CC
2		pncan3i.2	. . . 4 |- B e. CC
3	2	negcli 8541	. . 3 |- -u B e. CC
4	1, 3	negdii 8557	. 2 |- -u ( A + -u B ) = ( -u A + -u -u B )
5	1, 2	negsubi 8551	. . 3 |- ( A + -u B ) = ( A - B )
6	5	negeqi 8467	. 2 |- -u ( A + -u B ) = -u ( A - B )
7	2	negnegi 8543	. . 3 |- -u -u B = B
8	7	oveq2i 6061	. 2 |- ( -u A + -u -u B ) = ( -u A + B )
9	4, 6, 8	3eqtr3i 2261	1 |- -u ( A - B ) = ( -u A + B )

Syntax hints:   = wceq 1398   e. wcel 2203  (class class class)co 6050   CCcc 8125   + caddc 8130   - cmin 8444   -ucneg 8445

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-neg 8447

This theorem is referenced by:  negsubdi2i  8559  xnn0nnen  10799  resqrexlemover  11695