Theorem halfpm6th 8838

Description: One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)

Assertion

Ref	Expression
halfpm6th	|- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) )

Proof of Theorem halfpm6th

Step	Hyp	Ref	Expression
1		3cn 8699	. . . . . 6 |- 3 e. CC
2		ax-1cn 7632	. . . . . 6 |- 1 e. CC
3		2cn 8695	. . . . . 6 |- 2 e. CC
4		3re 8698	. . . . . . 7 |- 3 e. RR
5		3pos 8718	. . . . . . 7 |- 0 < 3
6	4, 5	gt0ap0ii 8302	. . . . . 6 |- 3 # 0
7		2ap0 8717	. . . . . 6 |- 2 # 0
8	1, 1, 2, 3, 6, 7	divmuldivapi 8439	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( ( 3 x. 1 ) / ( 3 x. 2 ) )
9	1, 6	dividapi 8412	. . . . . . 7 |- ( 3 / 3 ) = 1
10	9	oveq1i 5736	. . . . . 6 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 x. ( 1 / 2 ) )
11		halfcn 8832	. . . . . . 7 |- ( 1 / 2 ) e. CC
12	11	mulid2i 7687	. . . . . 6 |- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 )
13	10, 12	eqtri 2133	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 / 2 )
14	1	mulid1i 7686	. . . . . 6 |- ( 3 x. 1 ) = 3
15		3t2e6 8774	. . . . . 6 |- ( 3 x. 2 ) = 6
16	14, 15	oveq12i 5738	. . . . 5 |- ( ( 3 x. 1 ) / ( 3 x. 2 ) ) = ( 3 / 6 )
17	8, 13, 16	3eqtr3i 2141	. . . 4 |- ( 1 / 2 ) = ( 3 / 6 )
18	17	oveq1i 5736	. . 3 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
19		6cn 8706	. . . . 5 |- 6 e. CC
20		6re 8705	. . . . . 6 |- 6 e. RR
21		6pos 8725	. . . . . 6 |- 0 < 6
22	20, 21	gt0ap0ii 8302	. . . . 5 |- 6 # 0
23	19, 22	pm3.2i 268	. . . 4 |- ( 6 e. CC /\ 6 # 0 )
24		divsubdirap 8375	. . . 4 |- ( ( 3 e. CC /\ 1 e. CC /\ ( 6 e. CC /\ 6 # 0 ) ) -> ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) ) )
25	1, 2, 23, 24	mp3an 1296	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
26		3m1e2 8744	. . . . 5 |- ( 3 - 1 ) = 2
27	26	oveq1i 5736	. . . 4 |- ( ( 3 - 1 ) / 6 ) = ( 2 / 6 )
28	3	mulid2i 7687	. . . . 5 |- ( 1 x. 2 ) = 2
29	28, 15	oveq12i 5738	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 6 )
30	3, 7	dividapi 8412	. . . . . 6 |- ( 2 / 2 ) = 1
31	30	oveq2i 5737	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 / 3 ) x. 1 )
32	2, 1, 3, 3, 6, 7	divmuldivapi 8439	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 x. 2 ) / ( 3 x. 2 ) )
33	1, 6	recclapi 8409	. . . . . 6 |- ( 1 / 3 ) e. CC
34	33	mulid1i 7686	. . . . 5 |- ( ( 1 / 3 ) x. 1 ) = ( 1 / 3 )
35	31, 32, 34	3eqtr3i 2141	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 1 / 3 )
36	27, 29, 35	3eqtr2i 2139	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( 1 / 3 )
37	18, 25, 36	3eqtr2i 2139	. 2 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 )
38	1, 2, 19, 22	divdirapi 8436	. . . 4 |- ( ( 3 + 1 ) / 6 ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
39		df-4 8685	. . . . 5 |- 4 = ( 3 + 1 )
40	39	oveq1i 5736	. . . 4 |- ( 4 / 6 ) = ( ( 3 + 1 ) / 6 )
41	17	oveq1i 5736	. . . 4 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
42	38, 40, 41	3eqtr4ri 2144	. . 3 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 4 / 6 )
43		2t2e4 8772	. . . 4 |- ( 2 x. 2 ) = 4
44	43, 15	oveq12i 5738	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 4 / 6 )
45	30	oveq2i 5737	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 / 3 ) x. 1 )
46	3, 1, 3, 3, 6, 7	divmuldivapi 8439	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 x. 2 ) / ( 3 x. 2 ) )
47	3, 1, 6	divclapi 8421	. . . . 5 |- ( 2 / 3 ) e. CC
48	47	mulid1i 7686	. . . 4 |- ( ( 2 / 3 ) x. 1 ) = ( 2 / 3 )
49	45, 46, 48	3eqtr3i 2141	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 3 )
50	42, 44, 49	3eqtr2i 2139	. 2 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 )
51	37, 50	pm3.2i 268	1 |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) )

Syntax hints:   /\ wa 103   = wceq 1312   e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7539   0cc0 7541   1c1 7542   + caddc 7544   x. cmul 7546   - cmin 7850   # cap 8255   / cdiv 8339   2c2 8675   3c3 8676   4c4 8677   6c6 8679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-2 8683  df-3 8684  df-4 8685  df-5 8686  df-6 8687

This theorem is referenced by:  cos01bnd  11310