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Theorem pnpcan2 8200
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 8097 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  =  ( C  +  A ) )
213adant2 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  C )  =  ( C  +  A ) )
3 addcom 8097 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
433adant1 1015 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
52, 4oveq12d 5896 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( B  +  C ) )  =  ( ( C  +  A )  -  ( C  +  B
) ) )
6 pnpcan 8199 . . 3  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( C  +  A
)  -  ( C  +  B ) )  =  ( A  -  B ) )
763coml 1210 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  A
)  -  ( C  +  B ) )  =  ( A  -  B ) )
85, 7eqtrd 2210 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 978    = wceq 1353    e. wcel 2148  (class class class)co 5878   CCcc 7812    + caddc 7817    - cmin 8131
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-resscn 7906  ax-1cn 7907  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-sub 8133
This theorem is referenced by:  pnpcan2d  8309  addmodlteqALT  11868  omoe  11904
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