MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  8th4div3 Unicode version

Theorem 8th4div3 10116
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 8974 . . . 4  |-  1  e.  CC
2 8re 10003 . . . . 5  |-  8  e.  RR
32recni 9028 . . . 4  |-  8  e.  CC
4 4cn 9999 . . . 4  |-  4  e.  CC
5 3cn 9997 . . . 4  |-  3  e.  CC
6 8pos 10015 . . . . 5  |-  0  <  8
72, 6gt0ne0ii 9488 . . . 4  |-  8  =/=  0
8 3ne0 10010 . . . 4  |-  3  =/=  0
91, 3, 4, 5, 7, 8divmuldivi 9699 . . 3  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( ( 1  x.  4 )  /  (
8  x.  3 ) )
101, 4mulcomi 9022 . . . 4  |-  ( 1  x.  4 )  =  ( 4  x.  1 )
11 2cn 9995 . . . . . . . 8  |-  2  e.  CC
124, 11, 5mul32i 9187 . . . . . . 7  |-  ( ( 4  x.  2 )  x.  3 )  =  ( ( 4  x.  3 )  x.  2 )
13 4t2e8 10055 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
1413oveq1i 6023 . . . . . . 7  |-  ( ( 4  x.  2 )  x.  3 )  =  ( 8  x.  3 )
1512, 14eqtr3i 2402 . . . . . 6  |-  ( ( 4  x.  3 )  x.  2 )  =  ( 8  x.  3 )
164, 5, 11mulassi 9025 . . . . . 6  |-  ( ( 4  x.  3 )  x.  2 )  =  ( 4  x.  (
3  x.  2 ) )
1715, 16eqtr3i 2402 . . . . 5  |-  ( 8  x.  3 )  =  ( 4  x.  (
3  x.  2 ) )
18 3t2e6 10053 . . . . . 6  |-  ( 3  x.  2 )  =  6
1918oveq2i 6024 . . . . 5  |-  ( 4  x.  ( 3  x.  2 ) )  =  ( 4  x.  6 )
2017, 19eqtri 2400 . . . 4  |-  ( 8  x.  3 )  =  ( 4  x.  6 )
2110, 20oveq12i 6025 . . 3  |-  ( ( 1  x.  4 )  /  ( 8  x.  3 ) )  =  ( ( 4  x.  1 )  /  (
4  x.  6 ) )
229, 21eqtri 2400 . 2  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( ( 4  x.  1 )  /  (
4  x.  6 ) )
23 6re 10001 . . . 4  |-  6  e.  RR
2423recni 9028 . . 3  |-  6  e.  CC
25 6pos 10013 . . . 4  |-  0  <  6
2623, 25gt0ne0ii 9488 . . 3  |-  6  =/=  0
27 4re 9998 . . . 4  |-  4  e.  RR
28 4pos 10011 . . . 4  |-  0  <  4
2927, 28gt0ne0ii 9488 . . 3  |-  4  =/=  0
30 divcan5 9641 . . . 4  |-  ( ( 1  e.  CC  /\  ( 6  e.  CC  /\  6  =/=  0 )  /\  ( 4  e.  CC  /\  4  =/=  0 ) )  -> 
( ( 4  x.  1 )  /  (
4  x.  6 ) )  =  ( 1  /  6 ) )
311, 30mp3an1 1266 . . 3  |-  ( ( ( 6  e.  CC  /\  6  =/=  0 )  /\  ( 4  e.  CC  /\  4  =/=  0 ) )  -> 
( ( 4  x.  1 )  /  (
4  x.  6 ) )  =  ( 1  /  6 ) )
3224, 26, 4, 29, 31mp4an 655 . 2  |-  ( ( 4  x.  1 )  /  ( 4  x.  6 ) )  =  ( 1  /  6
)
3322, 32eqtri 2400 1  |-  ( ( 1  /  8 )  x.  ( 4  / 
3 ) )  =  ( 1  /  6
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    x. cmul 8921    / cdiv 9602   2c2 9974   3c3 9975   4c4 9976   6c6 9978   8c8 9980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989
  Copyright terms: Public domain W3C validator