Theorem halfpm6th 11253

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 11095	. . . . . 6 |- 3 e. CC
2		ax-1cn 9994	. . . . . 6 |- 1 e. CC
3		2cn 11091	. . . . . 6 |- 2 e. CC
4		3ne0 11115	. . . . . 6 |- 3 =/= 0
5		2ne0 11113	. . . . . 6 |- 2 =/= 0
6	1, 1, 2, 3, 4, 5	divmuldivi 10785	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( ( 3 x. 1 ) / ( 3 x. 2 ) )
7	1, 4	dividi 10758	. . . . . . 7 |- ( 3 / 3 ) = 1
8	7	oveq1i 6660	. . . . . 6 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 x. ( 1 / 2 ) )
9		halfcn 11247	. . . . . . 7 |- ( 1 / 2 ) e. CC
10	9	mulid2i 10043	. . . . . 6 |- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 )
11	8, 10	eqtri 2644	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 / 2 )
12	1	mulid1i 10042	. . . . . 6 |- ( 3 x. 1 ) = 3
13		3t2e6 11179	. . . . . 6 |- ( 3 x. 2 ) = 6
14	12, 13	oveq12i 6662	. . . . 5 |- ( ( 3 x. 1 ) / ( 3 x. 2 ) ) = ( 3 / 6 )
15	6, 11, 14	3eqtr3i 2652	. . . 4 |- ( 1 / 2 ) = ( 3 / 6 )
16	15	oveq1i 6660	. . 3 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
17		6cn 11102	. . . . 5 |- 6 e. CC
18		6re 11101	. . . . . 6 |- 6 e. RR
19		6pos 11119	. . . . . 6 |- 0 < 6
20	18, 19	gt0ne0ii 10564	. . . . 5 |- 6 =/= 0
21	17, 20	pm3.2i 471	. . . 4 |- ( 6 e. CC /\ 6 =/= 0 )
22		divsubdir 10721	. . . 4 |- ( ( 3 e. CC /\ 1 e. CC /\ ( 6 e. CC /\ 6 =/= 0 ) ) -> ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) ) )
23	1, 2, 21, 22	mp3an 1424	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
24		3m1e2 11137	. . . . 5 |- ( 3 - 1 ) = 2
25	24	oveq1i 6660	. . . 4 |- ( ( 3 - 1 ) / 6 ) = ( 2 / 6 )
26	3	mulid2i 10043	. . . . 5 |- ( 1 x. 2 ) = 2
27	26, 13	oveq12i 6662	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 6 )
28	3, 5	dividi 10758	. . . . . 6 |- ( 2 / 2 ) = 1
29	28	oveq2i 6661	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 / 3 ) x. 1 )
30	2, 1, 3, 3, 4, 5	divmuldivi 10785	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 x. 2 ) / ( 3 x. 2 ) )
31	1, 4	reccli 10755	. . . . . 6 |- ( 1 / 3 ) e. CC
32	31	mulid1i 10042	. . . . 5 |- ( ( 1 / 3 ) x. 1 ) = ( 1 / 3 )
33	29, 30, 32	3eqtr3i 2652	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 1 / 3 )
34	25, 27, 33	3eqtr2i 2650	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( 1 / 3 )
35	16, 23, 34	3eqtr2i 2650	. 2 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 )
36	1, 2, 17, 20	divdiri 10782	. . . 4 |- ( ( 3 + 1 ) / 6 ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
37		df-4 11081	. . . . 5 |- 4 = ( 3 + 1 )
38	37	oveq1i 6660	. . . 4 |- ( 4 / 6 ) = ( ( 3 + 1 ) / 6 )
39	15	oveq1i 6660	. . . 4 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
40	36, 38, 39	3eqtr4ri 2655	. . 3 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 4 / 6 )
41		2t2e4 11177	. . . 4 |- ( 2 x. 2 ) = 4
42	41, 13	oveq12i 6662	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 4 / 6 )
43	28	oveq2i 6661	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 / 3 ) x. 1 )
44	3, 1, 3, 3, 4, 5	divmuldivi 10785	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 x. 2 ) / ( 3 x. 2 ) )
45	3, 1, 4	divcli 10767	. . . . 5 |- ( 2 / 3 ) e. CC
46	45	mulid1i 10042	. . . 4 |- ( ( 2 / 3 ) x. 1 ) = ( 2 / 3 )
47	43, 44, 46	3eqtr3i 2652	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 3 )
48	40, 42, 47	3eqtr2i 2650	. 2 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 )
49	35, 48	pm3.2i 471	1 |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   3c3 11071   4c4 11072   6c6 11074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083

This theorem is referenced by:  cos01bnd  14916  sincos3rdpi  24268  1cubrlem  24568