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Theorem halfaddsub 9345
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 8388 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ A e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( A + A ) )
213anidm13 1330 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( A + A ) )
3 2times 9238 . . . . . 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 2265 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( 2 x. A ) )
65oveq1d 6016 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) + ( A - B ) ) / 2 ) = ( ( 2 x. A ) / 2 ) )
7 addcl 8124 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
8 subcl 8345 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
9 2cn 9181 . . . . . 6 |- 2 e. CC
10 2ap0 9203 . . . . . 6 |- 2 # 0
119, 10pm3.2i 272 . . . . 5 |- ( 2 e. CC /\ 2 # 0 )
12 divdirap 8844 . . . . 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 1360 . . . 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 8845 . . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( 2 x. A ) / 2 ) = A )
169, 10, 15mp3an23 1363 . . . 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 2270 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A )
19 pnncan 8387 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
20193anidm23 1331 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
21 2times 9238 . . . . . 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 2265 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( 2 x. B ) )
2423oveq1d 6016 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) - ( A - B ) ) / 2 ) = ( ( 2 x. B ) / 2 ) )
25 divsubdirap 8855 . . . . 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 1360 . . . 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 8845 . . . . 5 |- ( ( B e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( 2 x. B ) / 2 ) = B )
299, 10, 28mp3an23 1363 . . . 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 2270 . 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 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   + caddc 8002   x. cmul 8004   - cmin 8317   # cap 8728   / cdiv 8819   2c2 9161
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-2 9169
This theorem is referenced by:  addsin  12253  subsin  12254  addcos  12257  subcos  12258  ioo2bl  15225
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