Theorem negsubdii 8576

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 8559	. . 3 |- -u B e. CC
4	1, 3	negdii 8575	. 2 |- -u ( A + -u B ) = ( -u A + -u -u B )
5	1, 2	negsubi 8569	. . 3 |- ( A + -u B ) = ( A - B )
6	5	negeqi 8485	. 2 |- -u ( A + -u B ) = -u ( A - B )
7	2	negnegi 8561	. . 3 |- -u -u B = B
8	7	oveq2i 6070	. 2 |- ( -u A + -u -u B ) = ( -u A + B )
9	4, 6, 8	3eqtr3i 2263	1 |- -u ( A - B ) = ( -u A + B )

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6059   CCcc 8142   + caddc 8147   - cmin 8462   -ucneg 8463

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 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-setind 4665  ax-resscn 8236  ax-1cn 8237  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-addass 8246  ax-distr 8248  ax-i2m1 8249  ax-0id 8252  ax-rnegex 8253  ax-cnre 8255

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-iota 5318  df-fun 5360  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-sub 8464  df-neg 8465

This theorem is referenced by:  negsubdi2i  8577  xnn0nnen  10827  resqrexlemover  11725