ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  halfpm6th Unicode version

Theorem halfpm6th 9205
Description: One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
Assertion
Ref Expression
halfpm6th  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )

Proof of Theorem halfpm6th
StepHypRef Expression
1 3cn 9059 . . . . . 6  |-  3  e.  CC
2 ax-1cn 7967 . . . . . 6  |-  1  e.  CC
3 2cn 9055 . . . . . 6  |-  2  e.  CC
4 3re 9058 . . . . . . 7  |-  3  e.  RR
5 3pos 9078 . . . . . . 7  |-  0  <  3
64, 5gt0ap0ii 8649 . . . . . 6  |-  3 #  0
7 2ap0 9077 . . . . . 6  |-  2 #  0
81, 1, 2, 3, 6, 7divmuldivapi 8793 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
91, 6dividapi 8766 . . . . . . 7  |-  ( 3  /  3 )  =  1
109oveq1i 5929 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
11 halfcn 9199 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1211mullidi 8024 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1310, 12eqtri 2214 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
141mulid1i 8023 . . . . . 6  |-  ( 3  x.  1 )  =  3
15 3t2e6 9141 . . . . . 6  |-  ( 3  x.  2 )  =  6
1614, 15oveq12i 5931 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
178, 13, 163eqtr3i 2222 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1817oveq1i 5929 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
19 6cn 9066 . . . . 5  |-  6  e.  CC
20 6re 9065 . . . . . 6  |-  6  e.  RR
21 6pos 9085 . . . . . 6  |-  0  <  6
2220, 21gt0ap0ii 8649 . . . . 5  |-  6 #  0
2319, 22pm3.2i 272 . . . 4  |-  ( 6  e.  CC  /\  6 #  0 )
24 divsubdirap 8729 . . . 4  |-  ( ( 3  e.  CC  /\  1  e.  CC  /\  (
6  e.  CC  /\  6 #  0 ) )  -> 
( ( 3  -  1 )  /  6
)  =  ( ( 3  /  6 )  -  ( 1  / 
6 ) ) )
251, 2, 23, 24mp3an 1348 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
26 3m1e2 9104 . . . . 5  |-  ( 3  -  1 )  =  2
2726oveq1i 5929 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
283mullidi 8024 . . . . 5  |-  ( 1  x.  2 )  =  2
2928, 15oveq12i 5931 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
303, 7dividapi 8766 . . . . . 6  |-  ( 2  /  2 )  =  1
3130oveq2i 5930 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
322, 1, 3, 3, 6, 7divmuldivapi 8793 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
331, 6recclapi 8763 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3433mulid1i 8023 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3531, 32, 343eqtr3i 2222 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
3627, 29, 353eqtr2i 2220 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
3718, 25, 363eqtr2i 2220 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
381, 2, 19, 22divdirapi 8790 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
39 df-4 9045 . . . . 5  |-  4  =  ( 3  +  1 )
4039oveq1i 5929 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
4117oveq1i 5929 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4238, 40, 413eqtr4ri 2225 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
43 2t2e4 9139 . . . 4  |-  ( 2  x.  2 )  =  4
4443, 15oveq12i 5931 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4530oveq2i 5930 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
463, 1, 3, 3, 6, 7divmuldivapi 8793 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
473, 1, 6divclapi 8775 . . . . 5  |-  ( 2  /  3 )  e.  CC
4847mulid1i 8023 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
4945, 46, 483eqtr3i 2222 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
5042, 44, 493eqtr2i 2220 . 2  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
5137, 50pm3.2i 272 1  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    e. wcel 2164   class class class wbr 4030  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875    + caddc 7877    x. cmul 7879    - cmin 8192   # cap 8602    / cdiv 8693   2c2 9035   3c3 9036   4c4 9037   6c6 9039
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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047
This theorem is referenced by:  cos01bnd  11904
  Copyright terms: Public domain W3C validator