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Theorem halfaddsub 9166
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 8212 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ A e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( A + A ) )
213anidm13 1306 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( A + A ) )
3 2times 9060 . . . . . 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 2223 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( 2 x. A ) )
65oveq1d 5903 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) + ( A - B ) ) / 2 ) = ( ( 2 x. A ) / 2 ) )
7 addcl 7949 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
8 subcl 8169 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
9 2cn 9003 . . . . . 6 |- 2 e. CC
10 2ap0 9025 . . . . . 6 |- 2 # 0
119, 10pm3.2i 272 . . . . 5 |- ( 2 e. CC /\ 2 # 0 )
12 divdirap 8667 . . . . 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 1336 . . . 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 8668 . . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( 2 x. A ) / 2 ) = A )
169, 10, 15mp3an23 1339 . . . 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 2228 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A )
19 pnncan 8211 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
20193anidm23 1307 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
21 2times 9060 . . . . . 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 2223 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( 2 x. B ) )
2423oveq1d 5903 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) - ( A - B ) ) / 2 ) = ( ( 2 x. B ) / 2 ) )
25 divsubdirap 8678 . . . . 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 1336 . . . 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 8668 . . . . 5 |- ( ( B e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( 2 x. B ) / 2 ) = B )
299, 10, 28mp3an23 1339 . . . 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 2228 . 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 1363   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7822   0cc0 7824   + caddc 7827   x. cmul 7829   - cmin 8141   # cap 8551   / cdiv 8642   2c2 8983
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-2 8991
This theorem is referenced by:  addsin  11763  subsin  11764  addcos  11767  subcos  11768  ioo2bl  14314
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