ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xp1d2m1eqxm1d2 Unicode version

Theorem xp1d2m1eqxm1d2 9184
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 8105 . . . 4  |-  ( X  e.  CC  ->  ( X  +  1 )  e.  CC )
21halfcld 9176 . . 3  |-  ( X  e.  CC  ->  (
( X  +  1 )  /  2 )  e.  CC )
3 peano2cnm 8236 . . 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 8236 . . 3  |-  ( X  e.  CC  ->  ( X  -  1 )  e.  CC )
65halfcld 9176 . 2  |-  ( X  e.  CC  ->  (
( X  -  1 )  /  2 )  e.  CC )
7 2cnd 9005 . 2  |-  ( X  e.  CC  ->  2  e.  CC )
8 2ap0 9025 . . 3  |-  2 #  0
98a1i 9 . 2  |-  ( X  e.  CC  ->  2 #  0 )
10 1cnd 7986 . . . 4  |-  ( X  e.  CC  ->  1  e.  CC )
112, 10, 7subdird 8385 . . 3  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  -  1 )  x.  2 )  =  ( ( ( ( X  +  1 )  /  2 )  x.  2 )  -  ( 1  x.  2 ) ) )
121, 7, 9divcanap1d 8761 . . . 4  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  x.  2 )  =  ( X  + 
1 ) )
137mulid2d 7989 . . . 4  |-  ( X  e.  CC  ->  (
1  x.  2 )  =  2 )
1412, 13oveq12d 5906 . . 3  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( X  +  1 )  - 
2 ) )
155, 7, 9divcanap1d 8761 . . . 4  |-  ( X  e.  CC  ->  (
( ( X  - 
1 )  /  2
)  x.  2 )  =  ( X  - 
1 ) )
16 2m1e1 9050 . . . . . 6  |-  ( 2  -  1 )  =  1
1716a1i 9 . . . . 5  |-  ( X  e.  CC  ->  (
2  -  1 )  =  1 )
1817oveq2d 5904 . . . 4  |-  ( X  e.  CC  ->  ( X  -  ( 2  -  1 ) )  =  ( X  - 
1 ) )
19 id 19 . . . . 5  |-  ( X  e.  CC  ->  X  e.  CC )
2019, 7, 10subsub3d 8311 . . . 4  |-  ( X  e.  CC  ->  ( X  -  ( 2  -  1 ) )  =  ( ( X  +  1 )  - 
2 ) )
2115, 18, 203eqtr2rd 2227 . . 3  |-  ( X  e.  CC  ->  (
( X  +  1 )  -  2 )  =  ( ( ( X  -  1 )  /  2 )  x.  2 ) )
2211, 14, 213eqtrd 2224 . 2  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  -  1 )  x.  2 )  =  ( ( ( X  -  1 )  /  2 )  x.  2 ) )
234, 6, 7, 9, 22mulcanap2ad 8634 1  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  -  1 )  =  ( ( X  -  1 )  / 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825    + caddc 7827    x. cmul 7829    - cmin 8141   # cap 8551    / cdiv 8642   2c2 8983
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-2 8991
This theorem is referenced by:  zob  11909  nno  11924  nn0ob  11926
  Copyright terms: Public domain W3C validator