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Theorem ppncan 7997
Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
Assertion
Ref Expression
ppncan  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  ( C  -  B ) )  =  ( A  +  C ) )

Proof of Theorem ppncan
StepHypRef Expression
1 addcom 7892 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
213adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  B )  =  ( B  +  A ) )
32oveq1d 5782 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  ( B  -  C ) )  =  ( ( B  +  A )  -  ( B  -  C
) ) )
4 addcl 7738 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
543adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  B )  e.  CC )
6 subsub2 7983 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  ( B  -  C ) )  =  ( ( A  +  B )  +  ( C  -  B
) ) )
75, 6syld3an1 1262 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  ( B  -  C ) )  =  ( ( A  +  B )  +  ( C  -  B
) ) )
8 pnncan 7996 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  -  ( B  -  C ) )  =  ( A  +  C ) )
983com12 1185 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  -  ( B  -  C ) )  =  ( A  +  C ) )
103, 7, 93eqtr3d 2178 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  ( C  -  B ) )  =  ( A  +  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5767   CCcc 7611    + caddc 7616    - cmin 7926
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
This theorem is referenced by:  ppncand  8106  halfaddsub  8947  ptolemy  12894
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