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Theorem halfpm6th 9952
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 9834 . . . . . 6  |-  3  e.  CC
2 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
3 2cn 9832 . . . . . 6  |-  2  e.  CC
4 3ne0 9847 . . . . . 6  |-  3  =/=  0
5 2ne0 9845 . . . . . 6  |-  2  =/=  0
61, 1, 2, 3, 4, 5divmuldivi 9536 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( ( 3  x.  1 )  /  (
3  x.  2 ) )
71, 4dividi 9509 . . . . . . 7  |-  ( 3  /  3 )  =  1
87oveq1i 5884 . . . . . 6  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  x.  (
1  /  2 ) )
9 2re 9831 . . . . . . . . 9  |-  2  e.  RR
109, 5rereccli 9541 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
1110recni 8865 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1211mulid2i 8856 . . . . . 6  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
138, 12eqtri 2316 . . . . 5  |-  ( ( 3  /  3 )  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
141mulid1i 8855 . . . . . 6  |-  ( 3  x.  1 )  =  3
15 3t2e6 9888 . . . . . 6  |-  ( 3  x.  2 )  =  6
1614, 15oveq12i 5886 . . . . 5  |-  ( ( 3  x.  1 )  /  ( 3  x.  2 ) )  =  ( 3  /  6
)
176, 13, 163eqtr3i 2324 . . . 4  |-  ( 1  /  2 )  =  ( 3  /  6
)
1817oveq1i 5884 . . 3  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  -  (
1  /  6 ) )
19 6re 9838 . . . . . 6  |-  6  e.  RR
2019recni 8865 . . . . 5  |-  6  e.  CC
21 6pos 9850 . . . . . 6  |-  0  <  6
2219, 21gt0ne0ii 9325 . . . . 5  |-  6  =/=  0
2320, 22pm3.2i 441 . . . 4  |-  ( 6  e.  CC  /\  6  =/=  0 )
24 divsubdir 9472 . . . 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 9821 . . . . . . 7  |-  3  =  ( 2  +  1 )
2726oveq1i 5884 . . . . . 6  |-  ( 3  -  1 )  =  ( ( 2  +  1 )  -  1 )
28 pncan 9073 . . . . . . 7  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  +  1 )  -  1 )  =  2 )
293, 2, 28mp2an 653 . . . . . 6  |-  ( ( 2  +  1 )  -  1 )  =  2
3027, 29eqtri 2316 . . . . 5  |-  ( 3  -  1 )  =  2
3130oveq1i 5884 . . . 4  |-  ( ( 3  -  1 )  /  6 )  =  ( 2  /  6
)
323mulid2i 8856 . . . . 5  |-  ( 1  x.  2 )  =  2
3332, 15oveq12i 5886 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  6
)
343, 5dividi 9509 . . . . . 6  |-  ( 2  /  2 )  =  1
3534oveq2i 5885 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  / 
3 )  x.  1 )
362, 1, 3, 3, 4, 5divmuldivi 9536 . . . . 5  |-  ( ( 1  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 1  x.  2 )  /  (
3  x.  2 ) )
371, 4reccli 9506 . . . . . 6  |-  ( 1  /  3 )  e.  CC
3837mulid1i 8855 . . . . 5  |-  ( ( 1  /  3 )  x.  1 )  =  ( 1  /  3
)
3935, 36, 383eqtr3i 2324 . . . 4  |-  ( ( 1  x.  2 )  /  ( 3  x.  2 ) )  =  ( 1  /  3
)
4031, 33, 393eqtr2i 2322 . . 3  |-  ( ( 3  -  1 )  /  6 )  =  ( 1  /  3
)
4118, 25, 403eqtr2i 2322 . 2  |-  ( ( 1  /  2 )  -  ( 1  / 
6 ) )  =  ( 1  /  3
)
421, 2, 20, 22divdiri 9533 . . . 4  |-  ( ( 3  +  1 )  /  6 )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
43 df-4 9822 . . . . 5  |-  4  =  ( 3  +  1 )
4443oveq1i 5884 . . . 4  |-  ( 4  /  6 )  =  ( ( 3  +  1 )  /  6
)
4517oveq1i 5884 . . . 4  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( ( 3  / 
6 )  +  ( 1  /  6 ) )
4642, 44, 453eqtr4ri 2327 . . 3  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 4  /  6
)
47 2t2e4 9887 . . . 4  |-  ( 2  x.  2 )  =  4
4847, 15oveq12i 5886 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 4  /  6
)
4934oveq2i 5885 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  / 
3 )  x.  1 )
503, 1, 3, 3, 4, 5divmuldivi 9536 . . . 4  |-  ( ( 2  /  3 )  x.  ( 2  / 
2 ) )  =  ( ( 2  x.  2 )  /  (
3  x.  2 ) )
513, 1, 4divcli 9518 . . . . 5  |-  ( 2  /  3 )  e.  CC
5251mulid1i 8855 . . . 4  |-  ( ( 2  /  3 )  x.  1 )  =  ( 2  /  3
)
5349, 50, 523eqtr3i 2324 . . 3  |-  ( ( 2  x.  2 )  /  ( 3  x.  2 ) )  =  ( 2  /  3
)
5446, 48, 533eqtr2i 2322 . 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 1632    e. wcel 1696    =/= wne 2459  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   2c2 9811   3c3 9812   4c4 9813   6c6 9815
This theorem is referenced by:  cos01bnd  12482  sincos3rdpi  19900  1cubrlem  20153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824
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