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Theorem halfaddsub 9183
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 8229 . . . . . 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 9077 . . . . . 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 5911 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) + ( A - B ) ) / 2 ) = ( ( 2 x. A ) / 2 ) )
7 addcl 7966 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
8 subcl 8186 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
9 2cn 9020 . . . . . 6 |- 2 e. CC
10 2ap0 9042 . . . . . 6 |- 2 # 0
119, 10pm3.2i 272 . . . . 5 |- ( 2 e. CC /\ 2 # 0 )
12 divdirap 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 ) ) )
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 8685 . . . . 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 8228 . . . . . 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 9077 . . . . . 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 5911 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) - ( A - B ) ) / 2 ) = ( ( 2 x. B ) / 2 ) )
25 divsubdirap 8695 . . . . 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 8685 . . . . 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 5896   CCcc 7839   0cc0 7841   + caddc 7844   x. cmul 7846   - cmin 8158   # cap 8568   / cdiv 8659   2c2 9000
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 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
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 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-2 9008
This theorem is referenced by:  addsin  11782  subsin  11783  addcos  11786  subcos  11787  ioo2bl  14500
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