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Theorem pnpcan2 8227
Description: Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
Assertion
Ref Expression
pnpcan2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( B  +  C ) )  =  ( A  -  B ) )

Proof of Theorem pnpcan2
StepHypRef Expression
1 addcom 8124 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  =  ( C  +  A ) )
213adant2 1018 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  C )  =  ( C  +  A ) )
3 addcom 8124 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
433adant1 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
52, 4oveq12d 5914 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( B  +  C ) )  =  ( ( C  +  A )  -  ( C  +  B
) ) )
6 pnpcan 8226 . . 3  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( C  +  A
)  -  ( C  +  B ) )  =  ( A  -  B ) )
763coml 1212 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  A
)  -  ( C  +  B ) )  =  ( A  -  B ) )
85, 7eqtrd 2222 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( B  +  C ) )  =  ( A  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 980    = wceq 1364    e. wcel 2160  (class class class)co 5896   CCcc 7839    + caddc 7844    - cmin 8158
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sub 8160
This theorem is referenced by:  pnpcan2d  8336  addmodlteqALT  11897  omoe  11933
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