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Theorem halfaddsub 9172
Description: Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
Assertion
Ref Expression
halfaddsub  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  B )  /  2 )  +  ( ( A  -  B )  /  2
) )  =  A  /\  ( ( ( A  +  B )  /  2 )  -  ( ( A  -  B )  /  2
) )  =  B ) )

Proof of Theorem halfaddsub
StepHypRef Expression
1 ppncan 8218 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( A  +  B
)  +  ( A  -  B ) )  =  ( A  +  A ) )
213anidm13 1307 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( A  +  A ) )
3 2times 9066 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
43adantr 276 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
52, 4eqtr4d 2225 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
65oveq1d 5906 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  A )  /  2 ) )
7 addcl 7955 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 subcl 8175 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
9 2cn 9009 . . . . . 6  |-  2  e.  CC
10 2ap0 9031 . . . . . 6  |-  2 #  0
119, 10pm3.2i 272 . . . . 5  |-  ( 2  e.  CC  /\  2 #  0 )
12 divdirap 8673 . . . . 5  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC  /\  ( 2  e.  CC  /\  2 #  0 ) )  ->  ( ( ( A  +  B )  +  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )
1311, 12mp3an3 1337 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )
147, 8, 13syl2anc 411 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  +  ( ( A  -  B )  / 
2 ) ) )
15 divcanap3 8674 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
( 2  x.  A
)  /  2 )  =  A )
169, 10, 15mp3an23 1340 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  /  2 )  =  A )
1716adantr 276 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  /  2
)  =  A )
186, 14, 173eqtr3d 2230 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  A )
19 pnncan 8217 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
)  -  ( A  -  B ) )  =  ( B  +  B ) )
20193anidm23 1308 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( B  +  B ) )
21 2times 9066 . . . . . 6  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
2221adantl 277 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
2320, 22eqtr4d 2225 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( 2  x.  B ) )
2423oveq1d 5906 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  B )  /  2 ) )
25 divsubdirap 8684 . . . . 5  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC  /\  ( 2  e.  CC  /\  2 #  0 ) )  ->  ( ( ( A  +  B )  -  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  -  ( ( A  -  B )  /  2 ) ) )
2611, 25mp3an3 1337 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  -  ( ( A  -  B )  /  2 ) ) )
277, 8, 26syl2anc 411 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  -  ( ( A  -  B )  / 
2 ) ) )
28 divcanap3 8674 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
( 2  x.  B
)  /  2 )  =  B )
299, 10, 28mp3an23 1340 . . . 4  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  /  2 )  =  B )
3029adantl 277 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  /  2
)  =  B )
3124, 27, 303eqtr3d 2230 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  -  (
( A  -  B
)  /  2 ) )  =  B )
3218, 31jca 306 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  B )  /  2 )  +  ( ( A  -  B )  /  2
) )  =  A  /\  ( ( ( A  +  B )  /  2 )  -  ( ( A  -  B )  /  2
) )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7828   0cc0 7830    + caddc 7833    x. cmul 7835    - cmin 8147   # cap 8557    / cdiv 8648   2c2 8989
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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948
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-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  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 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-2 8997
This theorem is referenced by:  addsin  11769  subsin  11770  addcos  11773  subcos  11774  ioo2bl  14446
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