Theorem 8th4div3 9362

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 8124	. . . 4 |- 1 e. CC
2		8re 9227	. . . . 5 |- 8 e. RR
3	2	recni 8190	. . . 4 |- 8 e. CC
4		4cn 9220	. . . 4 |- 4 e. CC
5		3cn 9217	. . . 4 |- 3 e. CC
6		8pos 9245	. . . . 5 |- 0 < 8
7	2, 6	gt0ap0ii 8807	. . . 4 |- 8 # 0
8		3re 9216	. . . . 5 |- 3 e. RR
9		3pos 9236	. . . . 5 |- 0 < 3
10	8, 9	gt0ap0ii 8807	. . . 4 |- 3 # 0
11	1, 3, 4, 5, 7, 10	divmuldivapi 8951	. . 3 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 1 x. 4 ) / ( 8 x. 3 ) )
12	1, 4	mulcomi 8184	. . . 4 |- ( 1 x. 4 ) = ( 4 x. 1 )
13		2cn 9213	. . . . . . . 8 |- 2 e. CC
14	4, 13, 5	mul32i 8325	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( ( 4 x. 3 ) x. 2 )
15		4t2e8 9301	. . . . . . . 8 |- ( 4 x. 2 ) = 8
16	15	oveq1i 6027	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( 8 x. 3 )
17	14, 16	eqtr3i 2254	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 8 x. 3 )
18	4, 5, 13	mulassi 8187	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 4 x. ( 3 x. 2 ) )
19	17, 18	eqtr3i 2254	. . . . 5 |- ( 8 x. 3 ) = ( 4 x. ( 3 x. 2 ) )
20		3t2e6 9299	. . . . . 6 |- ( 3 x. 2 ) = 6
21	20	oveq2i 6028	. . . . 5 |- ( 4 x. ( 3 x. 2 ) ) = ( 4 x. 6 )
22	19, 21	eqtri 2252	. . . 4 |- ( 8 x. 3 ) = ( 4 x. 6 )
23	12, 22	oveq12i 6029	. . 3 |- ( ( 1 x. 4 ) / ( 8 x. 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
24	11, 23	eqtri 2252	. 2 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
25		6re 9223	. . . 4 |- 6 e. RR
26	25	recni 8190	. . 3 |- 6 e. CC
27		6pos 9243	. . . 4 |- 0 < 6
28	25, 27	gt0ap0ii 8807	. . 3 |- 6 # 0
29		4re 9219	. . . 4 |- 4 e. RR
30		4pos 9239	. . . 4 |- 0 < 4
31	29, 30	gt0ap0ii 8807	. . 3 |- 4 # 0
32		divcanap5 8893	. . . 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 1360	. . 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 2252	1 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   x. cmul 8036   # cap 8760   / cdiv 8851   2c2 9193   3c3 9194   4c4 9195   6c6 9197   8c8 9199

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207

This theorem is referenced by: (None)