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Theorem halfaddsub 8384
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 7469 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( A  +  B
)  +  ( A  -  B ) )  =  ( A  +  A ) )
213anidm13 1228 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( A  +  A ) )
3 2times 8279 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
43adantr 270 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
52, 4eqtr4d 2118 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
65oveq1d 5578 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  A )  /  2 ) )
7 addcl 7212 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 subcl 7426 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
9 2cn 8229 . . . . . 6  |-  2  e.  CC
10 2ap0 8251 . . . . . 6  |-  2 #  0
119, 10pm3.2i 266 . . . . 5  |-  ( 2  e.  CC  /\  2 #  0 )
12 divdirap 7904 . . . . 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 1258 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )
147, 8, 13syl2anc 403 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  +  ( ( A  -  B )  / 
2 ) ) )
15 divcanap3 7905 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
( 2  x.  A
)  /  2 )  =  A )
169, 10, 15mp3an23 1261 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  /  2 )  =  A )
1716adantr 270 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  /  2
)  =  A )
186, 14, 173eqtr3d 2123 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  A )
19 pnncan 7468 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
)  -  ( A  -  B ) )  =  ( B  +  B ) )
20193anidm23 1229 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( B  +  B ) )
21 2times 8279 . . . . . 6  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
2221adantl 271 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
2320, 22eqtr4d 2118 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( 2  x.  B ) )
2423oveq1d 5578 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  B )  /  2 ) )
25 divsubdirap 7915 . . . . 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 1258 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  -  ( ( A  -  B )  /  2 ) ) )
277, 8, 26syl2anc 403 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  -  ( ( A  -  B )  / 
2 ) ) )
28 divcanap3 7905 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
( 2  x.  B
)  /  2 )  =  B )
299, 10, 28mp3an23 1261 . . . 4  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  /  2 )  =  B )
3029adantl 271 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  /  2
)  =  B )
3124, 27, 303eqtr3d 2123 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  -  (
( A  -  B
)  /  2 ) )  =  B )
3218, 31jca 300 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 102    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093   0cc0 7095    + caddc 7098    x. cmul 7100    - cmin 7398   # cap 7800    / cdiv 7879   2c2 8208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-2 8217
This theorem is referenced by: (None)
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