Theorem halfpm6th 9141

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 8996	. . . . . 6 |- 3 e. CC
2		ax-1cn 7906	. . . . . 6 |- 1 e. CC
3		2cn 8992	. . . . . 6 |- 2 e. CC
4		3re 8995	. . . . . . 7 |- 3 e. RR
5		3pos 9015	. . . . . . 7 |- 0 < 3
6	4, 5	gt0ap0ii 8587	. . . . . 6 |- 3 # 0
7		2ap0 9014	. . . . . 6 |- 2 # 0
8	1, 1, 2, 3, 6, 7	divmuldivapi 8731	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( ( 3 x. 1 ) / ( 3 x. 2 ) )
9	1, 6	dividapi 8704	. . . . . . 7 |- ( 3 / 3 ) = 1
10	9	oveq1i 5887	. . . . . 6 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 x. ( 1 / 2 ) )
11		halfcn 9135	. . . . . . 7 |- ( 1 / 2 ) e. CC
12	11	mullidi 7962	. . . . . 6 |- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 )
13	10, 12	eqtri 2198	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 / 2 )
14	1	mulid1i 7961	. . . . . 6 |- ( 3 x. 1 ) = 3
15		3t2e6 9077	. . . . . 6 |- ( 3 x. 2 ) = 6
16	14, 15	oveq12i 5889	. . . . 5 |- ( ( 3 x. 1 ) / ( 3 x. 2 ) ) = ( 3 / 6 )
17	8, 13, 16	3eqtr3i 2206	. . . 4 |- ( 1 / 2 ) = ( 3 / 6 )
18	17	oveq1i 5887	. . 3 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
19		6cn 9003	. . . . 5 |- 6 e. CC
20		6re 9002	. . . . . 6 |- 6 e. RR
21		6pos 9022	. . . . . 6 |- 0 < 6
22	20, 21	gt0ap0ii 8587	. . . . 5 |- 6 # 0
23	19, 22	pm3.2i 272	. . . 4 |- ( 6 e. CC /\ 6 # 0 )
24		divsubdirap 8667	. . . 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 1337	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
26		3m1e2 9041	. . . . 5 |- ( 3 - 1 ) = 2
27	26	oveq1i 5887	. . . 4 |- ( ( 3 - 1 ) / 6 ) = ( 2 / 6 )
28	3	mullidi 7962	. . . . 5 |- ( 1 x. 2 ) = 2
29	28, 15	oveq12i 5889	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 6 )
30	3, 7	dividapi 8704	. . . . . 6 |- ( 2 / 2 ) = 1
31	30	oveq2i 5888	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 / 3 ) x. 1 )
32	2, 1, 3, 3, 6, 7	divmuldivapi 8731	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 x. 2 ) / ( 3 x. 2 ) )
33	1, 6	recclapi 8701	. . . . . 6 |- ( 1 / 3 ) e. CC
34	33	mulid1i 7961	. . . . 5 |- ( ( 1 / 3 ) x. 1 ) = ( 1 / 3 )
35	31, 32, 34	3eqtr3i 2206	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 1 / 3 )
36	27, 29, 35	3eqtr2i 2204	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( 1 / 3 )
37	18, 25, 36	3eqtr2i 2204	. 2 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 )
38	1, 2, 19, 22	divdirapi 8728	. . . 4 |- ( ( 3 + 1 ) / 6 ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
39		df-4 8982	. . . . 5 |- 4 = ( 3 + 1 )
40	39	oveq1i 5887	. . . 4 |- ( 4 / 6 ) = ( ( 3 + 1 ) / 6 )
41	17	oveq1i 5887	. . . 4 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
42	38, 40, 41	3eqtr4ri 2209	. . 3 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 4 / 6 )
43		2t2e4 9075	. . . 4 |- ( 2 x. 2 ) = 4
44	43, 15	oveq12i 5889	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 4 / 6 )
45	30	oveq2i 5888	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 / 3 ) x. 1 )
46	3, 1, 3, 3, 6, 7	divmuldivapi 8731	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 x. 2 ) / ( 3 x. 2 ) )
47	3, 1, 6	divclapi 8713	. . . . 5 |- ( 2 / 3 ) e. CC
48	47	mulid1i 7961	. . . 4 |- ( ( 2 / 3 ) x. 1 ) = ( 2 / 3 )
49	45, 46, 48	3eqtr3i 2206	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 3 )
50	42, 44, 49	3eqtr2i 2204	. 2 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 )
51	37, 50	pm3.2i 272	1 |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   - cmin 8130   # cap 8540   / cdiv 8631   2c2 8972   3c3 8973   4c4 8974   6c6 8976

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984

This theorem is referenced by:  cos01bnd  11768