ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  halfaddsub Unicode version
Theorem halfaddsub 9273
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 8316 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ A e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( A + A ) )
213anidm13 1309 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( A + A ) )
3 2times 9166 . . . . . 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 2241 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A - B ) ) = ( 2 x. A ) )
65oveq1d 5961 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) + ( A - B ) ) / 2 ) = ( ( 2 x. A ) / 2 ) )
7 addcl 8052 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
8 subcl 8273 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
9 2cn 9109 . . . . . 6 |- 2 e. CC
10 2ap0 9131 . . . . . 6 |- 2 # 0
119, 10pm3.2i 272 . . . . 5 |- ( 2 e. CC /\ 2 # 0 )
12 divdirap 8772 . . . . 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 1339 . . . 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 8773 . . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( 2 x. A ) / 2 ) = A )
169, 10, 15mp3an23 1342 . . . 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 2246 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A )
19 pnncan 8315 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
20193anidm23 1310 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
21 2times 9166 . . . . . 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 2241 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - ( A - B ) ) = ( 2 x. B ) )
2423oveq1d 5961 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) - ( A - B ) ) / 2 ) = ( ( 2 x. B ) / 2 ) )
25 divsubdirap 8783 . . . . 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 1339 . . . 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 8773 . . . . 5 |- ( ( B e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( 2 x. B ) / 2 ) = B )
299, 10, 28mp3an23 1342 . . . 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 2246 . 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 1373   e. wcel 2176   class class class wbr 4045  (class class class)co 5946   CCcc 7925   0cc0 7927   + caddc 7930   x. cmul 7932   - cmin 8245   # cap 8656   / cdiv 8747   2c2 9089
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-po 4344  df-iso 4345  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-2 9097
This theorem is referenced by:  addsin  12086  subsin  12087  addcos  12090  subcos  12091  ioo2bl  15056
  Copyright terms: Public domain W3C validator