Theorem 8th4div3 9031

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 7804	. . . 4 |- 1 e. CC
2		8re 8897	. . . . 5 |- 8 e. RR
3	2	recni 7869	. . . 4 |- 8 e. CC
4		4cn 8890	. . . 4 |- 4 e. CC
5		3cn 8887	. . . 4 |- 3 e. CC
6		8pos 8915	. . . . 5 |- 0 < 8
7	2, 6	gt0ap0ii 8482	. . . 4 |- 8 # 0
8		3re 8886	. . . . 5 |- 3 e. RR
9		3pos 8906	. . . . 5 |- 0 < 3
10	8, 9	gt0ap0ii 8482	. . . 4 |- 3 # 0
11	1, 3, 4, 5, 7, 10	divmuldivapi 8624	. . 3 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 1 x. 4 ) / ( 8 x. 3 ) )
12	1, 4	mulcomi 7863	. . . 4 |- ( 1 x. 4 ) = ( 4 x. 1 )
13		2cn 8883	. . . . . . . 8 |- 2 e. CC
14	4, 13, 5	mul32i 8001	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( ( 4 x. 3 ) x. 2 )
15		4t2e8 8970	. . . . . . . 8 |- ( 4 x. 2 ) = 8
16	15	oveq1i 5824	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( 8 x. 3 )
17	14, 16	eqtr3i 2177	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 8 x. 3 )
18	4, 5, 13	mulassi 7866	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 4 x. ( 3 x. 2 ) )
19	17, 18	eqtr3i 2177	. . . . 5 |- ( 8 x. 3 ) = ( 4 x. ( 3 x. 2 ) )
20		3t2e6 8968	. . . . . 6 |- ( 3 x. 2 ) = 6
21	20	oveq2i 5825	. . . . 5 |- ( 4 x. ( 3 x. 2 ) ) = ( 4 x. 6 )
22	19, 21	eqtri 2175	. . . 4 |- ( 8 x. 3 ) = ( 4 x. 6 )
23	12, 22	oveq12i 5826	. . 3 |- ( ( 1 x. 4 ) / ( 8 x. 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
24	11, 23	eqtri 2175	. 2 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
25		6re 8893	. . . 4 |- 6 e. RR
26	25	recni 7869	. . 3 |- 6 e. CC
27		6pos 8913	. . . 4 |- 0 < 6
28	25, 27	gt0ap0ii 8482	. . 3 |- 6 # 0
29		4re 8889	. . . 4 |- 4 e. RR
30		4pos 8909	. . . 4 |- 0 < 4
31	29, 30	gt0ap0ii 8482	. . 3 |- 4 # 0
32		divcanap5 8566	. . . 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 1303	. . 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 424	. 2 |- ( ( 4 x. 1 ) / ( 4 x. 6 ) ) = ( 1 / 6 )
35	24, 34	eqtri 2175	1 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 2125   class class class wbr 3961  (class class class)co 5814   CCcc 7709   0cc0 7711   1c1 7712   x. cmul 7716   # cap 8435   / cdiv 8524   2c2 8863   3c3 8864   4c4 8865   6c6 8867   8c8 8869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-id 4248  df-po 4251  df-iso 4252  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-2 8871  df-3 8872  df-4 8873  df-5 8874  df-6 8875  df-7 8876  df-8 8877

This theorem is referenced by: (None)