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Theorem halfpm6th 8569
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 8432 . . . . . 6  |-  3  e.  CC
2 ax-1cn 7382 . . . . . 6  |-  1  e.  CC
3 2cn 8428 . . . . . 6  |-  2  e.  CC
4 3re 8431 . . . . . . 7  |-  3  e.  RR
5 3pos 8451 . . . . . . 7  |-  0  <  3
64, 5gt0ap0ii 8045 . . . . . 6  |-  3 #  0
7 2ap0 8450 . . . . . 6  |-  2 #  0
81, 1, 2, 3, 6, 7divmuldivapi 8178 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
91, 6dividapi 8151 . . . . . . 7  |-  ( 3  /  3 )  =  1
109oveq1i 5623 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
11 halfcn 8563 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1211mulid2i 7435 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1310, 12eqtri 2105 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
141mulid1i 7434 . . . . . 6  |-  ( 3  x.  1 )  =  3
15 3t2e6 8506 . . . . . 6  |-  ( 3  x.  2 )  =  6
1614, 15oveq12i 5625 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
178, 13, 163eqtr3i 2113 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1817oveq1i 5623 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
19 6cn 8439 . . . . 5  |-  6  e.  CC
20 6re 8438 . . . . . 6  |-  6  e.  RR
21 6pos 8458 . . . . . 6  |-  0  <  6
2220, 21gt0ap0ii 8045 . . . . 5  |-  6 #  0
2319, 22pm3.2i 266 . . . 4  |-  ( 6  e.  CC  /\  6 #  0 )
24 divsubdirap 8114 . . . 4  |-  ( ( 3  e.  CC  /\  1  e.  CC  /\  (
6  e.  CC  /\  6 #  0 ) )  -> 
( ( 3  -  1 )  /  6
)  =  ( ( 3  /  6 )  -  ( 1  / 
6 ) ) )
251, 2, 23, 24mp3an 1271 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
26 3m1e2 8476 . . . . 5  |-  ( 3  -  1 )  =  2
2726oveq1i 5623 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
283mulid2i 7435 . . . . 5  |-  ( 1  x.  2 )  =  2
2928, 15oveq12i 5625 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
303, 7dividapi 8151 . . . . . 6  |-  ( 2  /  2 )  =  1
3130oveq2i 5624 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
322, 1, 3, 3, 6, 7divmuldivapi 8178 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
331, 6recclapi 8148 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3433mulid1i 7434 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3531, 32, 343eqtr3i 2113 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
3627, 29, 353eqtr2i 2111 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
3718, 25, 363eqtr2i 2111 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
381, 2, 19, 22divdirapi 8175 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
39 df-4 8418 . . . . 5  |-  4  =  ( 3  +  1 )
4039oveq1i 5623 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
4117oveq1i 5623 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4238, 40, 413eqtr4ri 2116 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
43 2t2e4 8504 . . . 4  |-  ( 2  x.  2 )  =  4
4443, 15oveq12i 5625 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4530oveq2i 5624 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
463, 1, 3, 3, 6, 7divmuldivapi 8178 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
473, 1, 6divclapi 8160 . . . . 5  |-  ( 2  /  3 )  e.  CC
4847mulid1i 7434 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
4945, 46, 483eqtr3i 2113 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
5042, 44, 493eqtr2i 2111 . 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 1287    e. wcel 1436   class class class wbr 3820  (class class class)co 5613   CCcc 7292   0cc0 7294   1c1 7295    + caddc 7297    x. cmul 7299    - cmin 7597   # cap 7999    / cdiv 8078   2c2 8407   3c3 8408   4c4 8409   6c6 8411
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-mulrcl 7388  ax-addcom 7389  ax-mulcom 7390  ax-addass 7391  ax-mulass 7392  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-1rid 7396  ax-0id 7397  ax-rnegex 7398  ax-precex 7399  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405  ax-pre-mulgt0 7406  ax-pre-mulext 7407
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-po 4097  df-iso 4098  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-iota 4946  df-fun 4983  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-reap 7993  df-ap 8000  df-div 8079  df-2 8416  df-3 8417  df-4 8418  df-5 8419  df-6 8420
This theorem is referenced by: (None)
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