Theorem xp1d2m1eqxm1d2 9325

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 8242	. . . 4 |- ( X e. CC -> ( X + 1 ) e. CC )
2	1	halfcld 9317	. . 3 |- ( X e. CC -> ( ( X + 1 ) / 2 ) e. CC )
3		peano2cnm 8373	. . 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 8373	. . 3 |- ( X e. CC -> ( X - 1 ) e. CC )
6	5	halfcld 9317	. 2 |- ( X e. CC -> ( ( X - 1 ) / 2 ) e. CC )
7		2cnd 9144	. 2 |- ( X e. CC -> 2 e. CC )
8		2ap0 9164	. . 3 |- 2 # 0
9	8	a1i 9	. 2 |- ( X e. CC -> 2 # 0 )
10		1cnd 8123	. . . 4 |- ( X e. CC -> 1 e. CC )
11	2, 10, 7	subdird 8522	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) )
12	1, 7, 9	divcanap1d 8899	. . . 4 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) x. 2 ) = ( X + 1 ) )
13	7	mulid2d 8126	. . . 4 |- ( X e. CC -> ( 1 x. 2 ) = 2 )
14	12, 13	oveq12d 5985	. . 3 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) x. 2 ) - ( 1 x. 2 ) ) = ( ( X + 1 ) - 2 ) )
15	5, 7, 9	divcanap1d 8899	. . . 4 |- ( X e. CC -> ( ( ( X - 1 ) / 2 ) x. 2 ) = ( X - 1 ) )
16		2m1e1 9189	. . . . . 6 |- ( 2 - 1 ) = 1
17	16	a1i 9	. . . . 5 |- ( X e. CC -> ( 2 - 1 ) = 1 )
18	17	oveq2d 5983	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( X - 1 ) )
19		id 19	. . . . 5 |- ( X e. CC -> X e. CC )
20	19, 7, 10	subsub3d 8448	. . . 4 |- ( X e. CC -> ( X - ( 2 - 1 ) ) = ( ( X + 1 ) - 2 ) )
21	15, 18, 20	3eqtr2rd 2247	. . 3 |- ( X e. CC -> ( ( X + 1 ) - 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
22	11, 14, 21	3eqtrd 2244	. 2 |- ( X e. CC -> ( ( ( ( X + 1 ) / 2 ) - 1 ) x. 2 ) = ( ( ( X - 1 ) / 2 ) x. 2 ) )
23	4, 6, 7, 9, 22	mulcanap2ad 8772	1 |- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   - cmin 8278   # cap 8689   / cdiv 8780   2c2 9122

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-2 9130

This theorem is referenced by:  zob  12317  nno  12332  nn0ob  12334