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Theorem ppncan 9861
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 9765 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
213adant3 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  B )  =  ( B  +  A ) )
32oveq1d 6299 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  ( B  -  C ) )  =  ( ( B  +  A )  -  ( B  -  C
) ) )
4 addcl 9574 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
543adant3 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  B )  e.  CC )
6 subsub2 9847 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  ( B  -  C ) )  =  ( ( A  +  B )  +  ( C  -  B
) ) )
75, 6syld3an1 1274 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  ( B  -  C ) )  =  ( ( A  +  B )  +  ( C  -  B
) ) )
8 pnncan 9860 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  -  ( B  -  C ) )  =  ( A  +  C ) )
983com12 1200 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  -  ( B  -  C ) )  =  ( A  +  C ) )
103, 7, 93eqtr3d 2516 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  ( C  -  B ) )  =  ( A  +  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6284   CCcc 9490    + caddc 9495    - cmin 9805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807
This theorem is referenced by:  ppncand  9970  halfaddsub  10772  pythagtriplem4  14202  pythagtriplem14  14211  ptolemy  22650  polid2i  25778  pellexlem2  30398
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