Theorem 8th4div3 9227

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 7989	. . . 4 |- 1 e. CC
2		8re 9092	. . . . 5 |- 8 e. RR
3	2	recni 8055	. . . 4 |- 8 e. CC
4		4cn 9085	. . . 4 |- 4 e. CC
5		3cn 9082	. . . 4 |- 3 e. CC
6		8pos 9110	. . . . 5 |- 0 < 8
7	2, 6	gt0ap0ii 8672	. . . 4 |- 8 # 0
8		3re 9081	. . . . 5 |- 3 e. RR
9		3pos 9101	. . . . 5 |- 0 < 3
10	8, 9	gt0ap0ii 8672	. . . 4 |- 3 # 0
11	1, 3, 4, 5, 7, 10	divmuldivapi 8816	. . 3 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 1 x. 4 ) / ( 8 x. 3 ) )
12	1, 4	mulcomi 8049	. . . 4 |- ( 1 x. 4 ) = ( 4 x. 1 )
13		2cn 9078	. . . . . . . 8 |- 2 e. CC
14	4, 13, 5	mul32i 8190	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( ( 4 x. 3 ) x. 2 )
15		4t2e8 9166	. . . . . . . 8 |- ( 4 x. 2 ) = 8
16	15	oveq1i 5935	. . . . . . 7 |- ( ( 4 x. 2 ) x. 3 ) = ( 8 x. 3 )
17	14, 16	eqtr3i 2219	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 8 x. 3 )
18	4, 5, 13	mulassi 8052	. . . . . 6 |- ( ( 4 x. 3 ) x. 2 ) = ( 4 x. ( 3 x. 2 ) )
19	17, 18	eqtr3i 2219	. . . . 5 |- ( 8 x. 3 ) = ( 4 x. ( 3 x. 2 ) )
20		3t2e6 9164	. . . . . 6 |- ( 3 x. 2 ) = 6
21	20	oveq2i 5936	. . . . 5 |- ( 4 x. ( 3 x. 2 ) ) = ( 4 x. 6 )
22	19, 21	eqtri 2217	. . . 4 |- ( 8 x. 3 ) = ( 4 x. 6 )
23	12, 22	oveq12i 5937	. . 3 |- ( ( 1 x. 4 ) / ( 8 x. 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
24	11, 23	eqtri 2217	. 2 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( ( 4 x. 1 ) / ( 4 x. 6 ) )
25		6re 9088	. . . 4 |- 6 e. RR
26	25	recni 8055	. . 3 |- 6 e. CC
27		6pos 9108	. . . 4 |- 0 < 6
28	25, 27	gt0ap0ii 8672	. . 3 |- 6 # 0
29		4re 9084	. . . 4 |- 4 e. RR
30		4pos 9104	. . . 4 |- 0 < 4
31	29, 30	gt0ap0ii 8672	. . 3 |- 4 # 0
32		divcanap5 8758	. . . 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 1335	. . 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 2217	1 |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   x. cmul 7901   # cap 8625   / cdiv 8716   2c2 9058   3c3 9059   4c4 9060   6c6 9062   8c8 9064

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072

This theorem is referenced by: (None)