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Theorem halfaddsub 10201
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 9343 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( A  +  B
)  +  ( A  -  B ) )  =  ( A  +  A ) )
213anidm13 1242 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( A  +  A ) )
3 2times 10099 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
43adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
52, 4eqtr4d 2471 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
65oveq1d 6096 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  A )  /  2 ) )
7 addcl 9072 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 subcl 9305 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
9 2cn 10070 . . . . . 6  |-  2  e.  CC
10 2ne0 10083 . . . . . 6  |-  2  =/=  0
119, 10pm3.2i 442 . . . . 5  |-  ( 2  e.  CC  /\  2  =/=  0 )
12 divdir 9701 . . . . 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 1268 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )
147, 8, 13syl2anc 643 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  +  ( ( A  -  B )  / 
2 ) ) )
15 divcan3 9702 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  A
)  /  2 )  =  A )
169, 10, 15mp3an23 1271 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  /  2 )  =  A )
1716adantr 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  /  2
)  =  A )
186, 14, 173eqtr3d 2476 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  A )
19 pnncan 9342 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
)  -  ( A  -  B ) )  =  ( B  +  B ) )
20193anidm23 1243 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( B  +  B ) )
21 2times 10099 . . . . . 6  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
2221adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
2320, 22eqtr4d 2471 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  B )
)  =  ( 2  x.  B ) )
2423oveq1d 6096 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( 2  x.  B )  /  2 ) )
25 divsubdir 9710 . . . . 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 1268 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B ) )  / 
2 )  =  ( ( ( A  +  B )  /  2
)  -  ( ( A  -  B )  /  2 ) ) )
277, 8, 26syl2anc 643 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  -  ( A  -  B
) )  /  2
)  =  ( ( ( A  +  B
)  /  2 )  -  ( ( A  -  B )  / 
2 ) ) )
28 divcan3 9702 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  B
)  /  2 )  =  B )
299, 10, 28mp3an23 1271 . . . 4  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  /  2 )  =  B )
3029adantl 453 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  /  2
)  =  B )
3124, 27, 303eqtr3d 2476 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  -  (
( A  -  B
)  /  2 ) )  =  B )
3218, 31jca 519 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599  (class class class)co 6081   CCcc 8988   0cc0 8990    + caddc 8993    x. cmul 8995    - cmin 9291    / cdiv 9677   2c2 10049
This theorem is referenced by:  addsin  12771  subsin  12772  addcos  12775  subcos  12776  ioo2bl  18824  dcubic  20686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058
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