Theorem halfpm6th 9460

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 9314	. . . . . 6 |- 3 e. CC
2		ax-1cn 8222	. . . . . 6 |- 1 e. CC
3		2cn 9310	. . . . . 6 |- 2 e. CC
4		3re 9313	. . . . . . 7 |- 3 e. RR
5		3pos 9333	. . . . . . 7 |- 0 < 3
6	4, 5	gt0ap0ii 8904	. . . . . 6 |- 3 # 0
7		2ap0 9332	. . . . . 6 |- 2 # 0
8	1, 1, 2, 3, 6, 7	divmuldivapi 9048	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( ( 3 x. 1 ) / ( 3 x. 2 ) )
9	1, 6	dividapi 9021	. . . . . . 7 |- ( 3 / 3 ) = 1
10	9	oveq1i 6062	. . . . . 6 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 x. ( 1 / 2 ) )
11		halfcn 9454	. . . . . . 7 |- ( 1 / 2 ) e. CC
12	11	mullidi 8279	. . . . . 6 |- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 )
13	10, 12	eqtri 2255	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 / 2 )
14	1	mulridi 8278	. . . . . 6 |- ( 3 x. 1 ) = 3
15		3t2e6 9396	. . . . . 6 |- ( 3 x. 2 ) = 6
16	14, 15	oveq12i 6064	. . . . 5 |- ( ( 3 x. 1 ) / ( 3 x. 2 ) ) = ( 3 / 6 )
17	8, 13, 16	3eqtr3i 2263	. . . 4 |- ( 1 / 2 ) = ( 3 / 6 )
18	17	oveq1i 6062	. . 3 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
19		6cn 9321	. . . . 5 |- 6 e. CC
20		6re 9320	. . . . . 6 |- 6 e. RR
21		6pos 9340	. . . . . 6 |- 0 < 6
22	20, 21	gt0ap0ii 8904	. . . . 5 |- 6 # 0
23	19, 22	pm3.2i 272	. . . 4 |- ( 6 e. CC /\ 6 # 0 )
24		divsubdirap 8984	. . . 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 1374	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
26		3m1e2 9359	. . . . 5 |- ( 3 - 1 ) = 2
27	26	oveq1i 6062	. . . 4 |- ( ( 3 - 1 ) / 6 ) = ( 2 / 6 )
28	3	mullidi 8279	. . . . 5 |- ( 1 x. 2 ) = 2
29	28, 15	oveq12i 6064	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 6 )
30	3, 7	dividapi 9021	. . . . . 6 |- ( 2 / 2 ) = 1
31	30	oveq2i 6063	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 / 3 ) x. 1 )
32	2, 1, 3, 3, 6, 7	divmuldivapi 9048	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 x. 2 ) / ( 3 x. 2 ) )
33	1, 6	recclapi 9018	. . . . . 6 |- ( 1 / 3 ) e. CC
34	33	mulridi 8278	. . . . 5 |- ( ( 1 / 3 ) x. 1 ) = ( 1 / 3 )
35	31, 32, 34	3eqtr3i 2263	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 1 / 3 )
36	27, 29, 35	3eqtr2i 2261	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( 1 / 3 )
37	18, 25, 36	3eqtr2i 2261	. 2 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 )
38	1, 2, 19, 22	divdirapi 9045	. . . 4 |- ( ( 3 + 1 ) / 6 ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
39		df-4 9300	. . . . 5 |- 4 = ( 3 + 1 )
40	39	oveq1i 6062	. . . 4 |- ( 4 / 6 ) = ( ( 3 + 1 ) / 6 )
41	17	oveq1i 6062	. . . 4 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
42	38, 40, 41	3eqtr4ri 2266	. . 3 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 4 / 6 )
43		2t2e4 9394	. . . 4 |- ( 2 x. 2 ) = 4
44	43, 15	oveq12i 6064	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 4 / 6 )
45	30	oveq2i 6063	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 / 3 ) x. 1 )
46	3, 1, 3, 3, 6, 7	divmuldivapi 9048	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 x. 2 ) / ( 3 x. 2 ) )
47	3, 1, 6	divclapi 9030	. . . . 5 |- ( 2 / 3 ) e. CC
48	47	mulridi 8278	. . . 4 |- ( ( 2 / 3 ) x. 1 ) = ( 2 / 3 )
49	45, 46, 48	3eqtr3i 2263	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 3 )
50	42, 44, 49	3eqtr2i 2261	. 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 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   - cmin 8446   # cap 8857   / cdiv 8948   2c2 9290   3c3 9291   4c4 9292   6c6 9294

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302

This theorem is referenced by:  cos01bnd  12448