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Theorem halfpm6th 10759
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 10609 . . . . . 6  |-  3  e.  CC
2 ax-1cn 9549 . . . . . 6  |-  1  e.  CC
3 2cn 10605 . . . . . 6  |-  2  e.  CC
4 3ne0 10629 . . . . . 6  |-  3  =/=  0
5 2ne0 10627 . . . . . 6  |-  2  =/=  0
61, 1, 2, 3, 4, 5divmuldivi 10303 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
71, 4dividi 10276 . . . . . . 7  |-  ( 3  /  3 )  =  1
87oveq1i 6293 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
9 halfcn 10754 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
109mulid2i 9598 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
118, 10eqtri 2496 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
121mulid1i 9597 . . . . . 6  |-  ( 3  x.  1 )  =  3
13 3t2e6 10686 . . . . . 6  |-  ( 3  x.  2 )  =  6
1412, 13oveq12i 6295 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
156, 11, 143eqtr3i 2504 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1615oveq1i 6293 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
17 6cn 10616 . . . . 5  |-  6  e.  CC
18 6re 10615 . . . . . 6  |-  6  e.  RR
19 6pos 10633 . . . . . 6  |-  0  <  6
2018, 19gt0ne0ii 10088 . . . . 5  |-  6  =/=  0
2117, 20pm3.2i 455 . . . 4  |-  ( 6  e.  CC  /\  6  =/=  0 )
22 divsubdir 10239 . . . 4  |-  ( ( 3  e.  CC  /\  1  e.  CC  /\  (
6  e.  CC  /\  6  =/=  0 ) )  ->  ( ( 3  -  1 )  / 
6 )  =  ( ( 3  /  6
)  -  ( 1  /  6 ) ) )
231, 2, 21, 22mp3an 1324 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
24 3m1e2 10651 . . . . 5  |-  ( 3  -  1 )  =  2
2524oveq1i 6293 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
263mulid2i 9598 . . . . 5  |-  ( 1  x.  2 )  =  2
2726, 13oveq12i 6295 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
283, 5dividi 10276 . . . . . 6  |-  ( 2  /  2 )  =  1
2928oveq2i 6294 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
302, 1, 3, 3, 4, 5divmuldivi 10303 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
311, 4reccli 10273 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3231mulid1i 9597 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3329, 30, 323eqtr3i 2504 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
3425, 27, 333eqtr2i 2502 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
3516, 23, 343eqtr2i 2502 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
361, 2, 17, 20divdiri 10300 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
37 df-4 10595 . . . . 5  |-  4  =  ( 3  +  1 )
3837oveq1i 6293 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
3915oveq1i 6293 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4036, 38, 393eqtr4ri 2507 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
41 2t2e4 10684 . . . 4  |-  ( 2  x.  2 )  =  4
4241, 13oveq12i 6295 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4328oveq2i 6294 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
443, 1, 3, 3, 4, 5divmuldivi 10303 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
453, 1, 4divcli 10285 . . . . 5  |-  ( 2  /  3 )  e.  CC
4645mulid1i 9597 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
4743, 44, 463eqtr3i 2504 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
4840, 42, 473eqtr2i 2502 . 2  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
4935, 48pm3.2i 455 1  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6283   CCcc 9489   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496    - cmin 9804    / cdiv 10205   2c2 10584   3c3 10585   4c4 10586   6c6 10588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597
This theorem is referenced by:  cos01bnd  13781  sincos3rdpi  22658  1cubrlem  22916
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