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

Theorem halfpm6th 8395
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 8258 . . . . . 6  |-  3  e.  CC
2 ax-1cn 7208 . . . . . 6  |-  1  e.  CC
3 2cn 8254 . . . . . 6  |-  2  e.  CC
4 3re 8257 . . . . . . 7  |-  3  e.  RR
5 3pos 8277 . . . . . . 7  |-  0  <  3
64, 5gt0ap0ii 7871 . . . . . 6  |-  3 #  0
7 2ap0 8276 . . . . . 6  |-  2 #  0
81, 1, 2, 3, 6, 7divmuldivapi 8004 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
91, 6dividapi 7977 . . . . . . 7  |-  ( 3  /  3 )  =  1
109oveq1i 5575 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
11 halfcn 8389 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1211mulid2i 7261 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1310, 12eqtri 2103 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
141mulid1i 7260 . . . . . 6  |-  ( 3  x.  1 )  =  3
15 3t2e6 8332 . . . . . 6  |-  ( 3  x.  2 )  =  6
1614, 15oveq12i 5577 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
178, 13, 163eqtr3i 2111 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1817oveq1i 5575 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
19 6cn 8265 . . . . 5  |-  6  e.  CC
20 6re 8264 . . . . . 6  |-  6  e.  RR
21 6pos 8284 . . . . . 6  |-  0  <  6
2220, 21gt0ap0ii 7871 . . . . 5  |-  6 #  0
2319, 22pm3.2i 266 . . . 4  |-  ( 6  e.  CC  /\  6 #  0 )
24 divsubdirap 7940 . . . 4  |-  ( ( 3  e.  CC  /\  1  e.  CC  /\  (
6  e.  CC  /\  6 #  0 ) )  -> 
( ( 3  -  1 )  /  6
)  =  ( ( 3  /  6 )  -  ( 1  / 
6 ) ) )
251, 2, 23, 24mp3an 1269 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
26 3m1e2 8302 . . . . 5  |-  ( 3  -  1 )  =  2
2726oveq1i 5575 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
283mulid2i 7261 . . . . 5  |-  ( 1  x.  2 )  =  2
2928, 15oveq12i 5577 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
303, 7dividapi 7977 . . . . . 6  |-  ( 2  /  2 )  =  1
3130oveq2i 5576 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
322, 1, 3, 3, 6, 7divmuldivapi 8004 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
331, 6recclapi 7974 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3433mulid1i 7260 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3531, 32, 343eqtr3i 2111 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
3627, 29, 353eqtr2i 2109 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
3718, 25, 363eqtr2i 2109 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
381, 2, 19, 22divdirapi 8001 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
39 df-4 8244 . . . . 5  |-  4  =  ( 3  +  1 )
4039oveq1i 5575 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
4117oveq1i 5575 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4238, 40, 413eqtr4ri 2114 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
43 2t2e4 8330 . . . 4  |-  ( 2  x.  2 )  =  4
4443, 15oveq12i 5577 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4530oveq2i 5576 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
463, 1, 3, 3, 6, 7divmuldivapi 8004 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
473, 1, 6divclapi 7986 . . . . 5  |-  ( 2  /  3 )  e.  CC
4847mulid1i 7260 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
4945, 46, 483eqtr3i 2111 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
5042, 44, 493eqtr2i 2109 . 2  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
5137, 50pm3.2i 266 1  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   class class class wbr 3806  (class class class)co 5565   CCcc 7118   0cc0 7120   1c1 7121    + caddc 7123    x. cmul 7125    - cmin 7423   # cap 7825    / cdiv 7904   2c2 8233   3c3 8234   4c4 8235   6c6 8237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-mulrcl 7214  ax-addcom 7215  ax-mulcom 7216  ax-addass 7217  ax-mulass 7218  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-1rid 7222  ax-0id 7223  ax-rnegex 7224  ax-precex 7225  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-apti 7230  ax-pre-ltadd 7231  ax-pre-mulgt0 7232  ax-pre-mulext 7233
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2613  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-reap 7819  df-ap 7826  df-div 7905  df-2 8242  df-3 8243  df-4 8244  df-5 8245  df-6 8246
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator