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Theorem 8th4div3 10755
Description: An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
Assertion
Ref Expression
8th4div3  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( 1  /  6
)

Proof of Theorem 8th4div3
StepHypRef Expression
1 ax-1cn 9539 . . . 4  |-  1  e.  CC
2 8re 10616 . . . . 5  |-  8  e.  RR
32recni 9597 . . . 4  |-  8  e.  CC
4 4cn 10609 . . . 4  |-  4  e.  CC
5 3cn 10606 . . . 4  |-  3  e.  CC
6 8pos 10632 . . . . 5  |-  0  <  8
72, 6gt0ne0ii 10085 . . . 4  |-  8  =/=  0
8 3ne0 10626 . . . 4  |-  3  =/=  0
91, 3, 4, 5, 7, 8divmuldivi 10300 . . 3  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( ( 1  x.  4 )  /  (
8  x.  3 ) )
101, 4mulcomi 9591 . . . 4  |-  ( 1  x.  4 )  =  ( 4  x.  1 )
11 2cn 10602 . . . . . . . 8  |-  2  e.  CC
124, 11, 5mul32i 9765 . . . . . . 7  |-  ( ( 4  x.  2 )  x.  3 )  =  ( ( 4  x.  3 )  x.  2 )
13 4t2e8 10685 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
1413oveq1i 6280 . . . . . . 7  |-  ( ( 4  x.  2 )  x.  3 )  =  ( 8  x.  3 )
1512, 14eqtr3i 2485 . . . . . 6  |-  ( ( 4  x.  3 )  x.  2 )  =  ( 8  x.  3 )
164, 5, 11mulassi 9594 . . . . . 6  |-  ( ( 4  x.  3 )  x.  2 )  =  ( 4  x.  (
3  x.  2 ) )
1715, 16eqtr3i 2485 . . . . 5  |-  ( 8  x.  3 )  =  ( 4  x.  (
3  x.  2 ) )
18 3t2e6 10683 . . . . . 6  |-  ( 3  x.  2 )  =  6
1918oveq2i 6281 . . . . 5  |-  ( 4  x.  ( 3  x.  2 ) )  =  ( 4  x.  6 )
2017, 19eqtri 2483 . . . 4  |-  ( 8  x.  3 )  =  ( 4  x.  6 )
2110, 20oveq12i 6282 . . 3  |-  ( ( 1  x.  4 )  /  ( 8  x.  3 ) )  =  ( ( 4  x.  1 )  /  (
4  x.  6 ) )
229, 21eqtri 2483 . 2  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( ( 4  x.  1 )  /  (
4  x.  6 ) )
23 6re 10612 . . . 4  |-  6  e.  RR
2423recni 9597 . . 3  |-  6  e.  CC
25 6pos 10630 . . . 4  |-  0  <  6
2623, 25gt0ne0ii 10085 . . 3  |-  6  =/=  0
27 4ne0 10628 . . 3  |-  4  =/=  0
28 divcan5 10242 . . . 4  |-  ( ( 1  e.  CC  /\  ( 6  e.  CC  /\  6  =/=  0 )  /\  ( 4  e.  CC  /\  4  =/=  0 ) )  -> 
( ( 4  x.  1 )  /  (
4  x.  6 ) )  =  ( 1  /  6 ) )
291, 28mp3an1 1309 . . 3  |-  ( ( ( 6  e.  CC  /\  6  =/=  0 )  /\  ( 4  e.  CC  /\  4  =/=  0 ) )  -> 
( ( 4  x.  1 )  /  (
4  x.  6 ) )  =  ( 1  /  6 ) )
3024, 26, 4, 27, 29mp4an 671 . 2  |-  ( ( 4  x.  1 )  /  ( 4  x.  6 ) )  =  ( 1  /  6
)
3122, 30eqtri 2483 1  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( 1  /  6
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    / cdiv 10202   2c2 10581   3c3 10582   4c4 10583   6c6 10585   8c8 10587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596
This theorem is referenced by: (None)
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