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Theorem halfaddsub 10094
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 9236 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( A  +  B
)  +  ( A  -  B ) )  =  ( A  +  A ) )
213anidm13 1241 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( A  +  A ) )
3 2times 9992 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
43adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
52, 4eqtr4d 2401 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
65oveq1d 5996 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  A )  /  2 ) )
7 addcl 8966 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 subcl 9198 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
9 2cn 9963 . . . . . 6  |-  2  e.  CC
10 2ne0 9976 . . . . . 6  |-  2  =/=  0
119, 10pm3.2i 441 . . . . 5  |-  ( 2  e.  CC  /\  2  =/=  0 )
12 divdir 9594 . . . . 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 1267 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )
147, 8, 13syl2anc 642 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  +  ( ( A  -  B )  / 
2 ) ) )
15 divcan3 9595 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  A
)  /  2 )  =  A )
169, 10, 15mp3an23 1270 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  /  2 )  =  A )
1716adantr 451 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  /  2
)  =  A )
186, 14, 173eqtr3d 2406 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  A )
19 pnncan 9235 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
)  -  ( A  -  B ) )  =  ( B  +  B ) )
20193anidm23 1242 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( B  +  B ) )
21 2times 9992 . . . . . 6  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
2221adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
2320, 22eqtr4d 2401 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( 2  x.  B ) )
2423oveq1d 5996 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  B )  /  2 ) )
25 divsubdir 9603 . . . . 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 1267 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  -  ( ( A  -  B )  /  2 ) ) )
277, 8, 26syl2anc 642 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  -  ( ( A  -  B )  / 
2 ) ) )
28 divcan3 9595 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  B
)  /  2 )  =  B )
299, 10, 28mp3an23 1270 . . . 4  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  /  2 )  =  B )
3029adantl 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  /  2
)  =  B )
3124, 27, 303eqtr3d 2406 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  -  (
( A  -  B
)  /  2 ) )  =  B )
3218, 31jca 518 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 358    = wceq 1647    e. wcel 1715    =/= wne 2529  (class class class)co 5981   CCcc 8882   0cc0 8884    + caddc 8887    x. cmul 8889    - cmin 9184    / cdiv 9570   2c2 9942
This theorem is referenced by:  addsin  12658  subsin  12659  addcos  12662  subcos  12663  ioo2bl  18512  dcubic  20364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-2 9951
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