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Theorem xp1d2m1eqxm1d2 9101
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
StepHypRef Expression
1 peano2cn 8025 . . . 4  |-  ( X  e.  CC  ->  ( X  +  1 )  e.  CC )
21halfcld 9093 . . 3  |-  ( X  e.  CC  ->  (
( X  +  1 )  /  2 )  e.  CC )
3 peano2cnm 8156 . . 3  |-  ( ( ( X  +  1 )  /  2 )  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  -  1 )  e.  CC )
42, 3syl 14 . 2  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  -  1 )  e.  CC )
5 peano2cnm 8156 . . 3  |-  ( X  e.  CC  ->  ( X  -  1 )  e.  CC )
65halfcld 9093 . 2  |-  ( X  e.  CC  ->  (
( X  -  1 )  /  2 )  e.  CC )
7 2cnd 8922 . 2  |-  ( X  e.  CC  ->  2  e.  CC )
8 2ap0 8942 . . 3  |-  2 #  0
98a1i 9 . 2  |-  ( X  e.  CC  ->  2 #  0 )
10 1cnd 7907 . . . 4  |-  ( X  e.  CC  ->  1  e.  CC )
112, 10, 7subdird 8305 . . 3  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  -  1 )  x.  2 )  =  ( ( ( ( X  +  1 )  /  2 )  x.  2 )  -  ( 1  x.  2 ) ) )
121, 7, 9divcanap1d 8679 . . . 4  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  x.  2 )  =  ( X  + 
1 ) )
137mulid2d 7909 . . . 4  |-  ( X  e.  CC  ->  (
1  x.  2 )  =  2 )
1412, 13oveq12d 5855 . . 3  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( X  +  1 )  - 
2 ) )
155, 7, 9divcanap1d 8679 . . . 4  |-  ( X  e.  CC  ->  (
( ( X  - 
1 )  /  2
)  x.  2 )  =  ( X  - 
1 ) )
16 2m1e1 8967 . . . . . 6  |-  ( 2  -  1 )  =  1
1716a1i 9 . . . . 5  |-  ( X  e.  CC  ->  (
2  -  1 )  =  1 )
1817oveq2d 5853 . . . 4  |-  ( X  e.  CC  ->  ( X  -  ( 2  -  1 ) )  =  ( X  - 
1 ) )
19 id 19 . . . . 5  |-  ( X  e.  CC  ->  X  e.  CC )
2019, 7, 10subsub3d 8231 . . . 4  |-  ( X  e.  CC  ->  ( X  -  ( 2  -  1 ) )  =  ( ( X  +  1 )  - 
2 ) )
2115, 18, 203eqtr2rd 2204 . . 3  |-  ( X  e.  CC  ->  (
( X  +  1 )  -  2 )  =  ( ( ( X  -  1 )  /  2 )  x.  2 ) )
2211, 14, 213eqtrd 2201 . 2  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  -  1 )  x.  2 )  =  ( ( ( X  -  1 )  /  2 )  x.  2 ) )
234, 6, 7, 9, 22mulcanap2ad 8553 1  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  -  1 )  =  ( ( X  -  1 )  / 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1342    e. wcel 2135   class class class wbr 3977  (class class class)co 5837   CCcc 7743   0cc0 7745   1c1 7746    + caddc 7748    x. cmul 7750    - cmin 8061   # cap 8471    / cdiv 8560   2c2 8900
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-1cn 7838  ax-1re 7839  ax-icn 7840  ax-addcl 7841  ax-addrcl 7842  ax-mulcl 7843  ax-mulrcl 7844  ax-addcom 7845  ax-mulcom 7846  ax-addass 7847  ax-mulass 7848  ax-distr 7849  ax-i2m1 7850  ax-0lt1 7851  ax-1rid 7852  ax-0id 7853  ax-rnegex 7854  ax-precex 7855  ax-cnre 7856  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860  ax-pre-ltadd 7861  ax-pre-mulgt0 7862  ax-pre-mulext 7863
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-po 4269  df-iso 4270  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-iota 5148  df-fun 5185  df-fv 5191  df-riota 5793  df-ov 5840  df-oprab 5841  df-mpo 5842  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-sub 8063  df-neg 8064  df-reap 8465  df-ap 8472  df-div 8561  df-2 8908
This theorem is referenced by:  zob  11817  nno  11832  nn0ob  11834
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