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Theorem negsubdii 8040
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
StepHypRef Expression
1 negidi.1 . . 3  |-  A  e.  CC
2 pncan3i.2 . . . 4  |-  B  e.  CC
32negcli 8023 . . 3  |-  -u B  e.  CC
41, 3negdii 8039 . 2  |-  -u ( A  +  -u B )  =  ( -u A  +  -u -u B )
51, 2negsubi 8033 . . 3  |-  ( A  +  -u B )  =  ( A  -  B
)
65negeqi 7949 . 2  |-  -u ( A  +  -u B )  =  -u ( A  -  B )
72negnegi 8025 . . 3  |-  -u -u B  =  B
87oveq2i 5778 . 2  |-  ( -u A  +  -u -u B
)  =  ( -u A  +  B )
94, 6, 83eqtr3i 2166 1  |-  -u ( A  -  B )  =  ( -u A  +  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5767   CCcc 7611    + caddc 7616    - cmin 7926   -ucneg 7927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-sub 7928  df-neg 7929
This theorem is referenced by:  negsubdi2i  8041  resqrexlemover  10775
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