Theorem xp1d2m1eqxm1d2 8283

Description: A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)

Assertion

Ref	Expression
xp1d2m1eqxm1d2	|- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )

Proof of Theorem xp1d2m1eqxm1d2

Step	Hyp	Ref	Expression
1		peano2cn 7243	. . . 4 |- ( X e. CC -> ( X + 1 ) e. CC )
2	1	halfcld 8275	. . 3 |- ( X e. CC -> ( ( X + 1 ) / 2 ) e. CC )
3		peano2cnm 7374	. . 3 |- ( ( ( X + 1 ) / 2 ) e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) e. CC )
4	2, 3	syl 14	. 2 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) e. CC )
5		peano2cnm 7374	. . 3 |- ( X e. CC -> ( X - 1 ) e. CC )
6	5	halfcld 8275	. 2 |- ( X e. CC -> ( ( X - 1 ) / 2 ) e. CC )
7		2cnd 8112	. 2 |- ( X e. CC -> 2 e. CC )
8		2ap0 8132	. . 3 |- 2 # 0
9	8	a1i 9	. 2 |- ( X e. CC -> 2 # 0 )
10		1cnd 7135	. . . 4 |- ( X e. CC -> 1 e. CC )
11	2, 10, 7	subdird 7519	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) )
12	1, 7, 9	divcanap1d 7878	. . . 4 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) x. 2 ) = ( X + 1 ) )
13	7	mulid2d 7137	. . . 4 |- ( X e. CC -> ( 1 x. 2 ) = 2 )
14	12, 13	oveq12d 5550	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) = ( ( X + 1 ) - 2 ) )
15	5, 7, 9	divcanap1d 7878	. . . 4 |- ( X e. CC -> ( ( ( X - 1 ) / 2 ) x. 2 ) = ( X - 1 ) )
16		2m1e1 8156	. . . . . 6 |- ( 2 - 1 ) = 1
17	16	a1i 9	. . . . 5 |- ( X e. CC -> ( 2 - 1 ) = 1 )
18	17	oveq2d 5548	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( X - 1 ) )
19		id 19	. . . . 5 |- ( X e. CC -> X e. CC )
20	19, 7, 10	subsub3d 7449	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( ( X + 1 ) - 2 ) )
21	15, 18, 20	3eqtr2rd 2120	. . 3 |- ( X e. CC -> ( ( X + 1 ) - 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
22	11, 14, 21	3eqtrd 2117	. 2 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
23	4, 6, 7, 9, 22	mulcanap2ad 7754	1 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   - cmin 7279   # cap 7681   / cdiv 7760   2c2 8089

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-2 8098

This theorem is referenced by:  zob  10291  nno  10306  nn0ob  10308