Theorem xp1d2m1eqxm1d2 9289

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 8206	. . . 4 |- ( X e. CC -> ( X + 1 ) e. CC )
2	1	halfcld 9281	. . 3 |- ( X e. CC -> ( ( X + 1 ) / 2 ) e. CC )
3		peano2cnm 8337	. . 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 8337	. . 3 |- ( X e. CC -> ( X - 1 ) e. CC )
6	5	halfcld 9281	. 2 |- ( X e. CC -> ( ( X - 1 ) / 2 ) e. CC )
7		2cnd 9108	. 2 |- ( X e. CC -> 2 e. CC )
8		2ap0 9128	. . 3 |- 2 # 0
9	8	a1i 9	. 2 |- ( X e. CC -> 2 # 0 )
10		1cnd 8087	. . . 4 |- ( X e. CC -> 1 e. CC )
11	2, 10, 7	subdird 8486	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) )
12	1, 7, 9	divcanap1d 8863	. . . 4 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) x. 2 ) = ( X + 1 ) )
13	7	mulid2d 8090	. . . 4 |- ( X e. CC -> ( 1 x. 2 ) = 2 )
14	12, 13	oveq12d 5961	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) = ( ( X + 1 ) - 2 ) )
15	5, 7, 9	divcanap1d 8863	. . . 4 |- ( X e. CC -> ( ( ( X - 1 ) / 2 ) x. 2 ) = ( X - 1 ) )
16		2m1e1 9153	. . . . . 6 |- ( 2 - 1 ) = 1
17	16	a1i 9	. . . . 5 |- ( X e. CC -> ( 2 - 1 ) = 1 )
18	17	oveq2d 5959	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( X - 1 ) )
19		id 19	. . . . 5 |- ( X e. CC -> X e. CC )
20	19, 7, 10	subsub3d 8412	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( ( X + 1 ) - 2 ) )
21	15, 18, 20	3eqtr2rd 2244	. . 3 |- ( X e. CC -> ( ( X + 1 ) - 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
22	11, 14, 21	3eqtrd 2241	. 2 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
23	4, 6, 7, 9, 22	mulcanap2ad 8736	1 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   - cmin 8242   # cap 8653   / cdiv 8744   2c2 9086

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-po 4342  df-iso 4343  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-2 9094

This theorem is referenced by:  zob  12173  nno  12188  nn0ob  12190