ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  halfaddsub Unicode version
Theorem halfaddsub 9225
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 8268 . . . . . 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 9118 . . . . . 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 2232 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( 2 x. A ) )
65oveq1d 5937 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) + ( A - B ) ) / 2 ) = ( ( 2 x. A ) / 2 ) )
7 addcl 8004 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
8 subcl 8225 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
9 2cn 9061 . . . . . 6 |- 2 e. CC
10 2ap0 9083 . . . . . 6 |- 2 # 0
119, 10pm3.2i 272 . . . . 5 |- ( 2 e. CC /\ 2 # 0 )
12 divdirap 8724 . . . . 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 8725 . . . . 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 2237 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A )
19 pnncan 8267 . . . . . 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 9118 . . . . . 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 2232 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( 2 x. B ) )
2423oveq1d 5937 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) - ( A - B ) ) / 2 ) = ( ( 2 x. B ) / 2 ) )
25 divsubdirap 8735 . . . . 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 8725 . . . . 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 2237 . 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 2167   class class class wbr 4033  (class class class)co 5922   CCcc 7877   0cc0 7879   + caddc 7882   x. cmul 7884   - cmin 8197   # cap 8608   / cdiv 8699   2c2 9041
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-2 9049
This theorem is referenced by:  addsin  11907  subsin  11908  addcos  11911  subcos  11912  ioo2bl  14787
  Copyright terms: Public domain W3C validator