Theorem 8th4div3 9353

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 8115	. . . 4 |- 1 e. CC
2		8re 9218	. . . . 5 |- 8 e. RR
3	2	recni 8181	. . . 4 |- 8 e. CC
4		4cn 9211	. . . 4 |- 4 e. CC
5		3cn 9208	. . . 4 |- 3 e. CC
6		8pos 9236	. . . . 5 |- 0 < 8
7	2, 6	gt0ap0ii 8798	. . . 4 |- 8 # 0
8		3re 9207	. . . . 5 |- 3 e. RR
9		3pos 9227	. . . . 5 |- 0 < 3
10	8, 9	gt0ap0ii 8798	. . . 4 |- 3 # 0
11	1, 3, 4, 5, 7, 10	divmuldivapi 8942	. . 3 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 1 x. 4 ) / ( 8 x. 3 ) )
12	1, 4	mulcomi 8175	. . . 4 |- ( 1 x. 4 ) = ( 4 x. 1 )
13		2cn 9204	. . . . . . . 8 |- 2 e. CC
14	4, 13, 5	mul32i 8316	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( ( 4 x. 3 ) x. 2 )
15		4t2e8 9292	. . . . . . . 8 |- ( 4 x. 2 ) = 8
16	15	oveq1i 6023	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( 8 x. 3 )
17	14, 16	eqtr3i 2252	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 8 x. 3 )
18	4, 5, 13	mulassi 8178	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 4 x. ( 3 x. 2 ) )
19	17, 18	eqtr3i 2252	. . . . 5 |- ( 8 x. 3 ) = ( 4 x. ( 3 x. 2 ) )
20		3t2e6 9290	. . . . . 6 |- ( 3 x. 2 ) = 6
21	20	oveq2i 6024	. . . . 5 |- ( 4 x. ( 3 x. 2 ) ) = ( 4 x. 6 )
22	19, 21	eqtri 2250	. . . 4 |- ( 8 x. 3 ) = ( 4 x. 6 )
23	12, 22	oveq12i 6025	. . 3 |- ( ( 1 x. 4 ) / ( 8 x. 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
24	11, 23	eqtri 2250	. 2 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
25		6re 9214	. . . 4 |- 6 e. RR
26	25	recni 8181	. . 3 |- 6 e. CC
27		6pos 9234	. . . 4 |- 0 < 6
28	25, 27	gt0ap0ii 8798	. . 3 |- 6 # 0
29		4re 9210	. . . 4 |- 4 e. RR
30		4pos 9230	. . . 4 |- 0 < 4
31	29, 30	gt0ap0ii 8798	. . 3 |- 4 # 0
32		divcanap5 8884	. . . 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 1358	. . 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 2250	1 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   0cc0 8022   1c1 8023   x. cmul 8027   # cap 8751   / cdiv 8842   2c2 9184   3c3 9185   4c4 9186   6c6 9188   8c8 9190

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198

This theorem is referenced by: (None)