Theorem xp1d2m1eqxm1d2 8996

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 7921	. . . 4 |- ( X e. CC -> ( X + 1 ) e. CC )
2	1	halfcld 8988	. . 3 |- ( X e. CC -> ( ( X + 1 ) / 2 ) e. CC )
3		peano2cnm 8052	. . 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 8052	. . 3 |- ( X e. CC -> ( X - 1 ) e. CC )
6	5	halfcld 8988	. 2 |- ( X e. CC -> ( ( X - 1 ) / 2 ) e. CC )
7		2cnd 8817	. 2 |- ( X e. CC -> 2 e. CC )
8		2ap0 8837	. . 3 |- 2 # 0
9	8	a1i 9	. 2 |- ( X e. CC -> 2 # 0 )
10		1cnd 7806	. . . 4 |- ( X e. CC -> 1 e. CC )
11	2, 10, 7	subdird 8201	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) )
12	1, 7, 9	divcanap1d 8575	. . . 4 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) x. 2 ) = ( X + 1 ) )
13	7	mulid2d 7808	. . . 4 |- ( X e. CC -> ( 1 x. 2 ) = 2 )
14	12, 13	oveq12d 5800	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) = ( ( X + 1 ) - 2 ) )
15	5, 7, 9	divcanap1d 8575	. . . 4 |- ( X e. CC -> ( ( ( X - 1 ) / 2 ) x. 2 ) = ( X - 1 ) )
16		2m1e1 8862	. . . . . 6 |- ( 2 - 1 ) = 1
17	16	a1i 9	. . . . 5 |- ( X e. CC -> ( 2 - 1 ) = 1 )
18	17	oveq2d 5798	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( X - 1 ) )
19		id 19	. . . . 5 |- ( X e. CC -> X e. CC )
20	19, 7, 10	subsub3d 8127	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( ( X + 1 ) - 2 ) )
21	15, 18, 20	3eqtr2rd 2180	. . 3 |- ( X e. CC -> ( ( X + 1 ) - 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
22	11, 14, 21	3eqtrd 2177	. 2 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
23	4, 6, 7, 9, 22	mulcanap2ad 8449	1 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   - cmin 7957   # cap 8367   / cdiv 8456   2c2 8795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-2 8803

This theorem is referenced by:  zob  11624  nno  11639  nn0ob  11641