Theorem 8th4div3 9351

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 8113	. . . 4 |- 1 e. CC
2		8re 9216	. . . . 5 |- 8 e. RR
3	2	recni 8179	. . . 4 |- 8 e. CC
4		4cn 9209	. . . 4 |- 4 e. CC
5		3cn 9206	. . . 4 |- 3 e. CC
6		8pos 9234	. . . . 5 |- 0 < 8
7	2, 6	gt0ap0ii 8796	. . . 4 |- 8 # 0
8		3re 9205	. . . . 5 |- 3 e. RR
9		3pos 9225	. . . . 5 |- 0 < 3
10	8, 9	gt0ap0ii 8796	. . . 4 |- 3 # 0
11	1, 3, 4, 5, 7, 10	divmuldivapi 8940	. . 3 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 1 x. 4 ) / ( 8 x. 3 ) )
12	1, 4	mulcomi 8173	. . . 4 |- ( 1 x. 4 ) = ( 4 x. 1 )
13		2cn 9202	. . . . . . . 8 |- 2 e. CC
14	4, 13, 5	mul32i 8314	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( ( 4 x. 3 ) x. 2 )
15		4t2e8 9290	. . . . . . . 8 |- ( 4 x. 2 ) = 8
16	15	oveq1i 6021	. . . . . . 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 8176	. . . . . 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 9288	. . . . . 6 |- ( 3 x. 2 ) = 6
21	20	oveq2i 6022	. . . . 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 6023	. . 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 9212	. . . 4 |- 6 e. RR
26	25	recni 8179	. . 3 |- 6 e. CC
27		6pos 9232	. . . 4 |- 0 < 6
28	25, 27	gt0ap0ii 8796	. . 3 |- 6 # 0
29		4re 9208	. . . 4 |- 4 e. RR
30		4pos 9228	. . . 4 |- 0 < 4
31	29, 30	gt0ap0ii 8796	. . 3 |- 4 # 0
32		divcanap5 8882	. . . 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 4084  (class class class)co 6011   CCcc 8018   0cc0 8020   1c1 8021   x. cmul 8025   # cap 8749   / cdiv 8840   2c2 9182   3c3 9183   4c4 9184   6c6 9186   8c8 9188

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 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137  ax-pre-mulext 8138

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 3890  df-br 4085  df-opab 4147  df-id 4386  df-po 4389  df-iso 4390  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-iota 5282  df-fun 5324  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-div 8841  df-2 9190  df-3 9191  df-4 9192  df-5 9193  df-6 9194  df-7 9195  df-8 9196

This theorem is referenced by: (None)