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Theorem halfpm6th 10542
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 10392 . . . . . 6  |-  3  e.  CC
2 ax-1cn 9336 . . . . . 6  |-  1  e.  CC
3 2cn 10388 . . . . . 6  |-  2  e.  CC
4 3ne0 10412 . . . . . 6  |-  3  =/=  0
5 2ne0 10410 . . . . . 6  |-  2  =/=  0
61, 1, 2, 3, 4, 5divmuldivi 10087 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
71, 4dividi 10060 . . . . . . 7  |-  ( 3  /  3 )  =  1
87oveq1i 6100 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
9 halfcn 10537 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
109mulid2i 9385 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
118, 10eqtri 2461 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
121mulid1i 9384 . . . . . 6  |-  ( 3  x.  1 )  =  3
13 3t2e6 10469 . . . . . 6  |-  ( 3  x.  2 )  =  6
1412, 13oveq12i 6102 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
156, 11, 143eqtr3i 2469 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1615oveq1i 6100 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
17 6cn 10399 . . . . 5  |-  6  e.  CC
18 6re 10398 . . . . . 6  |-  6  e.  RR
19 6pos 10416 . . . . . 6  |-  0  <  6
2018, 19gt0ne0ii 9872 . . . . 5  |-  6  =/=  0
2117, 20pm3.2i 452 . . . 4  |-  ( 6  e.  CC  /\  6  =/=  0 )
22 divsubdir 10023 . . . 4  |-  ( ( 3  e.  CC  /\  1  e.  CC  /\  (
6  e.  CC  /\  6  =/=  0 ) )  ->  ( ( 3  -  1 )  / 
6 )  =  ( ( 3  /  6
)  -  ( 1  /  6 ) ) )
231, 2, 21, 22mp3an 1309 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
24 3m1e2 10434 . . . . 5  |-  ( 3  -  1 )  =  2
2524oveq1i 6100 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
263mulid2i 9385 . . . . 5  |-  ( 1  x.  2 )  =  2
2726, 13oveq12i 6102 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
283, 5dividi 10060 . . . . . 6  |-  ( 2  /  2 )  =  1
2928oveq2i 6101 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
302, 1, 3, 3, 4, 5divmuldivi 10087 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
311, 4reccli 10057 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3231mulid1i 9384 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3329, 30, 323eqtr3i 2469 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
3425, 27, 333eqtr2i 2467 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
3516, 23, 343eqtr2i 2467 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
361, 2, 17, 20divdiri 10084 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
37 df-4 10378 . . . . 5  |-  4  =  ( 3  +  1 )
3837oveq1i 6100 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
3915oveq1i 6100 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4036, 38, 393eqtr4ri 2472 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
41 2t2e4 10467 . . . 4  |-  ( 2  x.  2 )  =  4
4241, 13oveq12i 6102 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4328oveq2i 6101 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
443, 1, 3, 3, 4, 5divmuldivi 10087 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
453, 1, 4divcli 10069 . . . . 5  |-  ( 2  /  3 )  e.  CC
4645mulid1i 9384 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
4743, 44, 463eqtr3i 2469 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
4840, 42, 473eqtr2i 2467 . 2  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
4935, 48pm3.2i 452 1  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    - cmin 9591    / cdiv 9989   2c2 10367   3c3 10368   4c4 10369   6c6 10371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380
This theorem is referenced by:  cos01bnd  13466  sincos3rdpi  21937  1cubrlem  22195
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