Theorem halfpm6th 9202

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 9057	. . . . . 6 |- 3 e. CC
2		ax-1cn 7965	. . . . . 6 |- 1 e. CC
3		2cn 9053	. . . . . 6 |- 2 e. CC
4		3re 9056	. . . . . . 7 |- 3 e. RR
5		3pos 9076	. . . . . . 7 |- 0 < 3
6	4, 5	gt0ap0ii 8647	. . . . . 6 |- 3 # 0
7		2ap0 9075	. . . . . 6 |- 2 # 0
8	1, 1, 2, 3, 6, 7	divmuldivapi 8791	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( ( 3 x. 1 ) / ( 3 x. 2 ) )
9	1, 6	dividapi 8764	. . . . . . 7 |- ( 3 / 3 ) = 1
10	9	oveq1i 5928	. . . . . 6 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 x. ( 1 / 2 ) )
11		halfcn 9196	. . . . . . 7 |- ( 1 / 2 ) e. CC
12	11	mullidi 8022	. . . . . 6 |- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 )
13	10, 12	eqtri 2214	. . . . 5 |- ( ( 3 / 3 ) x. ( 1 / 2 ) ) = ( 1 / 2 )
14	1	mulid1i 8021	. . . . . 6 |- ( 3 x. 1 ) = 3
15		3t2e6 9138	. . . . . 6 |- ( 3 x. 2 ) = 6
16	14, 15	oveq12i 5930	. . . . 5 |- ( ( 3 x. 1 ) / ( 3 x. 2 ) ) = ( 3 / 6 )
17	8, 13, 16	3eqtr3i 2222	. . . 4 |- ( 1 / 2 ) = ( 3 / 6 )
18	17	oveq1i 5928	. . 3 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
19		6cn 9064	. . . . 5 |- 6 e. CC
20		6re 9063	. . . . . 6 |- 6 e. RR
21		6pos 9083	. . . . . 6 |- 0 < 6
22	20, 21	gt0ap0ii 8647	. . . . 5 |- 6 # 0
23	19, 22	pm3.2i 272	. . . 4 |- ( 6 e. CC /\ 6 # 0 )
24		divsubdirap 8727	. . . 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 1348	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( ( 3 / 6 ) - ( 1 / 6 ) )
26		3m1e2 9102	. . . . 5 |- ( 3 - 1 ) = 2
27	26	oveq1i 5928	. . . 4 |- ( ( 3 - 1 ) / 6 ) = ( 2 / 6 )
28	3	mullidi 8022	. . . . 5 |- ( 1 x. 2 ) = 2
29	28, 15	oveq12i 5930	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 6 )
30	3, 7	dividapi 8764	. . . . . 6 |- ( 2 / 2 ) = 1
31	30	oveq2i 5929	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 / 3 ) x. 1 )
32	2, 1, 3, 3, 6, 7	divmuldivapi 8791	. . . . 5 |- ( ( 1 / 3 ) x. ( 2 / 2 ) ) = ( ( 1 x. 2 ) / ( 3 x. 2 ) )
33	1, 6	recclapi 8761	. . . . . 6 |- ( 1 / 3 ) e. CC
34	33	mulid1i 8021	. . . . 5 |- ( ( 1 / 3 ) x. 1 ) = ( 1 / 3 )
35	31, 32, 34	3eqtr3i 2222	. . . 4 |- ( ( 1 x. 2 ) / ( 3 x. 2 ) ) = ( 1 / 3 )
36	27, 29, 35	3eqtr2i 2220	. . 3 |- ( ( 3 - 1 ) / 6 ) = ( 1 / 3 )
37	18, 25, 36	3eqtr2i 2220	. 2 |- ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 )
38	1, 2, 19, 22	divdirapi 8788	. . . 4 |- ( ( 3 + 1 ) / 6 ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
39		df-4 9043	. . . . 5 |- 4 = ( 3 + 1 )
40	39	oveq1i 5928	. . . 4 |- ( 4 / 6 ) = ( ( 3 + 1 ) / 6 )
41	17	oveq1i 5928	. . . 4 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( ( 3 / 6 ) + ( 1 / 6 ) )
42	38, 40, 41	3eqtr4ri 2225	. . 3 |- ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 4 / 6 )
43		2t2e4 9136	. . . 4 |- ( 2 x. 2 ) = 4
44	43, 15	oveq12i 5930	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 4 / 6 )
45	30	oveq2i 5929	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 / 3 ) x. 1 )
46	3, 1, 3, 3, 6, 7	divmuldivapi 8791	. . . 4 |- ( ( 2 / 3 ) x. ( 2 / 2 ) ) = ( ( 2 x. 2 ) / ( 3 x. 2 ) )
47	3, 1, 6	divclapi 8773	. . . . 5 |- ( 2 / 3 ) e. CC
48	47	mulid1i 8021	. . . 4 |- ( ( 2 / 3 ) x. 1 ) = ( 2 / 3 )
49	45, 46, 48	3eqtr3i 2222	. . 3 |- ( ( 2 x. 2 ) / ( 3 x. 2 ) ) = ( 2 / 3 )
50	42, 44, 49	3eqtr2i 2220	. 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 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   # cap 8600   / cdiv 8691   2c2 9033   3c3 9034   4c4 9035   6c6 9037

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045

This theorem is referenced by:  cos01bnd  11901