Theorem 8th4div3 9155

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

Step	Hyp	Ref	Expression
1		ax-1cn 7921	. . . 4 |- 1 e. CC
2		8re 9021	. . . . 5 |- 8 e. RR
3	2	recni 7986	. . . 4 |- 8 e. CC
4		4cn 9014	. . . 4 |- 4 e. CC
5		3cn 9011	. . . 4 |- 3 e. CC
6		8pos 9039	. . . . 5 |- 0 < 8
7	2, 6	gt0ap0ii 8602	. . . 4 |- 8 # 0
8		3re 9010	. . . . 5 |- 3 e. RR
9		3pos 9030	. . . . 5 |- 0 < 3
10	8, 9	gt0ap0ii 8602	. . . 4 |- 3 # 0
11	1, 3, 4, 5, 7, 10	divmuldivapi 8746	. . 3 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 1 x. 4 ) / ( 8 x. 3 ) )
12	1, 4	mulcomi 7980	. . . 4 |- ( 1 x. 4 ) = ( 4 x. 1 )
13		2cn 9007	. . . . . . . 8 |- 2 e. CC
14	4, 13, 5	mul32i 8121	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( ( 4 x. 3 ) x. 2 )
15		4t2e8 9094	. . . . . . . 8 |- ( 4 x. 2 ) = 8
16	15	oveq1i 5900	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( 8 x. 3 )
17	14, 16	eqtr3i 2211	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 8 x. 3 )
18	4, 5, 13	mulassi 7983	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 4 x. ( 3 x. 2 ) )
19	17, 18	eqtr3i 2211	. . . . 5 |- ( 8 x. 3 ) = ( 4 x. ( 3 x. 2 ) )
20		3t2e6 9092	. . . . . 6 |- ( 3 x. 2 ) = 6
21	20	oveq2i 5901	. . . . 5 |- ( 4 x. ( 3 x. 2 ) ) = ( 4 x. 6 )
22	19, 21	eqtri 2209	. . . 4 |- ( 8 x. 3 ) = ( 4 x. 6 )
23	12, 22	oveq12i 5902	. . 3 |- ( ( 1 x. 4 ) / ( 8 x. 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
24	11, 23	eqtri 2209	. 2 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
25		6re 9017	. . . 4 |- 6 e. RR
26	25	recni 7986	. . 3 |- 6 e. CC
27		6pos 9037	. . . 4 |- 0 < 6
28	25, 27	gt0ap0ii 8602	. . 3 |- 6 # 0
29		4re 9013	. . . 4 |- 4 e. RR
30		4pos 9033	. . . 4 |- 0 < 4
31	29, 30	gt0ap0ii 8602	. . 3 |- 4 # 0
32		divcanap5 8688	. . . 4 |- ( ( 1 e. CC /\ ( 6 e. CC /\ 6 # 0 ) /\ ( 4 e. CC /\ 4 # 0 ) ) -> ( ( 4 x. 1 ) / ( 4 x. 6 ) ) = ( 1 / 6 ) )
33	1, 32	mp3an1 1334	. . 3 |- ( ( ( 6 e. CC /\ 6 # 0 ) /\ ( 4 e. CC /\ 4 # 0 ) ) -> ( ( 4 x. 1 ) / ( 4 x. 6 ) ) = ( 1 / 6 ) )
34	26, 28, 4, 31, 33	mp4an 427	. 2 |- ( ( 4 x. 1 ) / ( 4 x. 6 ) ) = ( 1 / 6 )
35	24, 34	eqtri 2209	1 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )

Syntax hints:   /\ wa 104   = wceq 1363   e. wcel 2159   class class class wbr 4017  (class class class)co 5890   CCcc 7826   0cc0 7828   1c1 7829   x. cmul 7833   # cap 8555   / cdiv 8646   2c2 8987   3c3 8988   4c4 8989   6c6 8991   8c8 8993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-iota 5192  df-fun 5232  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-2 8995  df-3 8996  df-4 8997  df-5 8998  df-6 8999  df-7 9000  df-8 9001

This theorem is referenced by: (None)