Theorem halfpm6th 9077

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 8932	. . . . . 6 |- 3 e. CC
2		ax-1cn 7846	. . . . . 6 |- 1 e. CC
3		2cn 8928	. . . . . 6 |- 2 e. CC
4		3re 8931	. . . . . . 7 |- 3 e. RR
5		3pos 8951	. . . . . . 7 |- 0 < 3
6	4, 5	gt0ap0ii 8526	. . . . . 6 |- 3 # 0
7		2ap0 8950	. . . . . 6 |- 2 # 0
8	1, 1, 2, 3, 6, 7	divmuldivapi 8668	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( ( 3 x. 1 ) / ( 3 x. 2 ) )
9	1, 6	dividapi 8641	. . . . . . 7 |- ( 3 / 3 ) = 1
10	9	oveq1i 5852	. . . . . 6 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 x. ( 1 / 2 ) )
11		halfcn 9071	. . . . . . 7 |- ( 1 / 2 ) e. CC
12	11	mulid2i 7902	. . . . . 6 |- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 )
13	10, 12	eqtri 2186	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 / 2 )
14	1	mulid1i 7901	. . . . . 6 |- ( 3 x. 1 ) = 3
15		3t2e6 9013	. . . . . 6 |- ( 3 x. 2 ) = 6
16	14, 15	oveq12i 5854	. . . . 5 |- ( ( 3 x. 1 ) / ( 3 x. 2 ) ) = ( 3 / 6 )
17	8, 13, 16	3eqtr3i 2194	. . . 4 |- ( 1 / 2 ) = ( 3 / 6 )
18	17	oveq1i 5852	. . 3 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
19		6cn 8939	. . . . 5 |- 6 e. CC
20		6re 8938	. . . . . 6 |- 6 e. RR
21		6pos 8958	. . . . . 6 |- 0 < 6
22	20, 21	gt0ap0ii 8526	. . . . 5 |- 6 # 0
23	19, 22	pm3.2i 270	. . . 4 |- ( 6 e. CC /\ 6 # 0 )
24		divsubdirap 8604	. . . 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 1327	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
26		3m1e2 8977	. . . . 5 |- ( 3 - 1 ) = 2
27	26	oveq1i 5852	. . . 4 |- ( ( 3 - 1 ) / 6 ) = ( 2 / 6 )
28	3	mulid2i 7902	. . . . 5 |- ( 1 x. 2 ) = 2
29	28, 15	oveq12i 5854	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 6 )
30	3, 7	dividapi 8641	. . . . . 6 |- ( 2 / 2 ) = 1
31	30	oveq2i 5853	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 / 3 ) x. 1 )
32	2, 1, 3, 3, 6, 7	divmuldivapi 8668	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 x. 2 ) / ( 3 x. 2 ) )
33	1, 6	recclapi 8638	. . . . . 6 |- ( 1 / 3 ) e. CC
34	33	mulid1i 7901	. . . . 5 |- ( ( 1 / 3 ) x. 1 ) = ( 1 / 3 )
35	31, 32, 34	3eqtr3i 2194	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 1 / 3 )
36	27, 29, 35	3eqtr2i 2192	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( 1 / 3 )
37	18, 25, 36	3eqtr2i 2192	. 2 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 )
38	1, 2, 19, 22	divdirapi 8665	. . . 4 |- ( ( 3 + 1 ) / 6 ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
39		df-4 8918	. . . . 5 |- 4 = ( 3 + 1 )
40	39	oveq1i 5852	. . . 4 |- ( 4 / 6 ) = ( ( 3 + 1 ) / 6 )
41	17	oveq1i 5852	. . . 4 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
42	38, 40, 41	3eqtr4ri 2197	. . 3 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 4 / 6 )
43		2t2e4 9011	. . . 4 |- ( 2 x. 2 ) = 4
44	43, 15	oveq12i 5854	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 4 / 6 )
45	30	oveq2i 5853	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 / 3 ) x. 1 )
46	3, 1, 3, 3, 6, 7	divmuldivapi 8668	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 x. 2 ) / ( 3 x. 2 ) )
47	3, 1, 6	divclapi 8650	. . . . 5 |- ( 2 / 3 ) e. CC
48	47	mulid1i 7901	. . . 4 |- ( ( 2 / 3 ) x. 1 ) = ( 2 / 3 )
49	45, 46, 48	3eqtr3i 2194	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 3 )
50	42, 44, 49	3eqtr2i 2192	. 2 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 )
51	37, 50	pm3.2i 270	1 |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   - cmin 8069   # cap 8479   / cdiv 8568   2c2 8908   3c3 8909   4c4 8910   6c6 8912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920

This theorem is referenced by:  cos01bnd  11699