Theorem xp1d2m1eqxm1d2 8722

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 7671	. . . 4 |- ( X e. CC -> ( X + 1 ) e. CC )
2	1	halfcld 8714	. . 3 |- ( X e. CC -> ( ( X + 1 ) / 2 ) e. CC )
3		peano2cnm 7802	. . 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 7802	. . 3 |- ( X e. CC -> ( X - 1 ) e. CC )
6	5	halfcld 8714	. 2 |- ( X e. CC -> ( ( X - 1 ) / 2 ) e. CC )
7		2cnd 8549	. 2 |- ( X e. CC -> 2 e. CC )
8		2ap0 8569	. . 3 |- 2 # 0
9	8	a1i 9	. 2 |- ( X e. CC -> 2 # 0 )
10		1cnd 7558	. . . 4 |- ( X e. CC -> 1 e. CC )
11	2, 10, 7	subdird 7947	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) )
12	1, 7, 9	divcanap1d 8312	. . . 4 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) x. 2 ) = ( X + 1 ) )
13	7	mulid2d 7560	. . . 4 |- ( X e. CC -> ( 1 x. 2 ) = 2 )
14	12, 13	oveq12d 5684	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) = ( ( X + 1 ) - 2 ) )
15	5, 7, 9	divcanap1d 8312	. . . 4 |- ( X e. CC -> ( ( ( X - 1 ) / 2 ) x. 2 ) = ( X - 1 ) )
16		2m1e1 8594	. . . . . 6 |- ( 2 - 1 ) = 1
17	16	a1i 9	. . . . 5 |- ( X e. CC -> ( 2 - 1 ) = 1 )
18	17	oveq2d 5682	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( X - 1 ) )
19		id 19	. . . . 5 |- ( X e. CC -> X e. CC )
20	19, 7, 10	subsub3d 7877	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( ( X + 1 ) - 2 ) )
21	15, 18, 20	3eqtr2rd 2128	. . 3 |- ( X e. CC -> ( ( X + 1 ) - 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
22	11, 14, 21	3eqtrd 2125	. 2 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
23	4, 6, 7, 9, 22	mulcanap2ad 8187	1 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439   class class class wbr 3851  (class class class)co 5666   CCcc 7402   0cc0 7404   1c1 7405   + caddc 7407   x. cmul 7409   - cmin 7707   # cap 8112   / cdiv 8193   2c2 8527

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-2 8535

This theorem is referenced by:  zob  11223  nno  11238  nn0ob  11240