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Theorem halfaddsub 8954
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 8004 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( A  +  B
)  +  ( A  -  B ) )  =  ( A  +  A ) )
213anidm13 1274 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( A  +  A ) )
3 2times 8848 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
43adantr 274 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
52, 4eqtr4d 2175 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
65oveq1d 5789 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  A )  /  2 ) )
7 addcl 7745 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 subcl 7961 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
9 2cn 8791 . . . . . 6  |-  2  e.  CC
10 2ap0 8813 . . . . . 6  |-  2 #  0
119, 10pm3.2i 270 . . . . 5  |-  ( 2  e.  CC  /\  2 #  0 )
12 divdirap 8457 . . . . 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 1304 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )
147, 8, 13syl2anc 408 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  +  ( ( A  -  B )  / 
2 ) ) )
15 divcanap3 8458 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
( 2  x.  A
)  /  2 )  =  A )
169, 10, 15mp3an23 1307 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  /  2 )  =  A )
1716adantr 274 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  /  2
)  =  A )
186, 14, 173eqtr3d 2180 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  A )
19 pnncan 8003 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
)  -  ( A  -  B ) )  =  ( B  +  B ) )
20193anidm23 1275 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( B  +  B ) )
21 2times 8848 . . . . . 6  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
2221adantl 275 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
2320, 22eqtr4d 2175 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( 2  x.  B ) )
2423oveq1d 5789 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  B )  /  2 ) )
25 divsubdirap 8468 . . . . 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 1304 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  -  ( ( A  -  B )  /  2 ) ) )
277, 8, 26syl2anc 408 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  -  ( ( A  -  B )  / 
2 ) ) )
28 divcanap3 8458 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
( 2  x.  B
)  /  2 )  =  B )
299, 10, 28mp3an23 1307 . . . 4  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  /  2 )  =  B )
3029adantl 275 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  /  2
)  =  B )
3124, 27, 303eqtr3d 2180 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  -  (
( A  -  B
)  /  2 ) )  =  B )
3218, 31jca 304 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 103    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   0cc0 7620    + caddc 7623    x. cmul 7625    - cmin 7933   # cap 8343    / cdiv 8432   2c2 8771
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-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-2 8779
This theorem is referenced by:  addsin  11449  subsin  11450  addcos  11453  subcos  11454  ioo2bl  12712
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