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Theorem halfpm6th 9936
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 9818 . . . . . 6  |-  3  e.  CC
2 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
3 2cn 9816 . . . . . 6  |-  2  e.  CC
4 3ne0 9831 . . . . . 6  |-  3  =/=  0
5 2ne0 9829 . . . . . 6  |-  2  =/=  0
61, 1, 2, 3, 4, 5divmuldivi 9520 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
71, 4dividi 9493 . . . . . . 7  |-  ( 3  /  3 )  =  1
87oveq1i 5868 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
9 2re 9815 . . . . . . . . 9  |-  2  e.  RR
109, 5rereccli 9525 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
1110recni 8849 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1211mulid2i 8840 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
138, 12eqtri 2303 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
141mulid1i 8839 . . . . . 6  |-  ( 3  x.  1 )  =  3
15 3t2e6 9872 . . . . . 6  |-  ( 3  x.  2 )  =  6
1614, 15oveq12i 5870 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
176, 13, 163eqtr3i 2311 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1817oveq1i 5868 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
19 6re 9822 . . . . . 6  |-  6  e.  RR
2019recni 8849 . . . . 5  |-  6  e.  CC
21 6pos 9834 . . . . . 6  |-  0  <  6
2219, 21gt0ne0ii 9309 . . . . 5  |-  6  =/=  0
2320, 22pm3.2i 441 . . . 4  |-  ( 6  e.  CC  /\  6  =/=  0 )
24 divsubdir 9456 . . . 4  |-  ( ( 3  e.  CC  /\  1  e.  CC  /\  (
6  e.  CC  /\  6  =/=  0 ) )  ->  ( ( 3  -  1 )  / 
6 )  =  ( ( 3  /  6
)  -  ( 1  /  6 ) ) )
251, 2, 23, 24mp3an 1277 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
26 df-3 9805 . . . . . . 7  |-  3  =  ( 2  +  1 )
2726oveq1i 5868 . . . . . 6  |-  ( 3  -  1 )  =  ( ( 2  +  1 )  -  1 )
28 pncan 9057 . . . . . . 7  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  +  1 )  -  1 )  =  2 )
293, 2, 28mp2an 653 . . . . . 6  |-  ( ( 2  +  1 )  -  1 )  =  2
3027, 29eqtri 2303 . . . . 5  |-  ( 3  -  1 )  =  2
3130oveq1i 5868 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
323mulid2i 8840 . . . . 5  |-  ( 1  x.  2 )  =  2
3332, 15oveq12i 5870 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
343, 5dividi 9493 . . . . . 6  |-  ( 2  /  2 )  =  1
3534oveq2i 5869 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
362, 1, 3, 3, 4, 5divmuldivi 9520 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
371, 4reccli 9490 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3837mulid1i 8839 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3935, 36, 383eqtr3i 2311 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
4031, 33, 393eqtr2i 2309 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
4118, 25, 403eqtr2i 2309 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
421, 2, 20, 22divdiri 9517 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
43 df-4 9806 . . . . 5  |-  4  =  ( 3  +  1 )
4443oveq1i 5868 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
4517oveq1i 5868 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4642, 44, 453eqtr4ri 2314 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
47 2t2e4 9871 . . . 4  |-  ( 2  x.  2 )  =  4
4847, 15oveq12i 5870 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4934oveq2i 5869 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
503, 1, 3, 3, 4, 5divmuldivi 9520 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
513, 1, 4divcli 9502 . . . . 5  |-  ( 2  /  3 )  e.  CC
5251mulid1i 8839 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
5349, 50, 523eqtr3i 2311 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
5446, 48, 533eqtr2i 2309 . 2  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
5541, 54pm3.2i 441 1  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   6c6 9799
This theorem is referenced by:  cos01bnd  12466  sincos3rdpi  19884  1cubrlem  20137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808
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