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

Theorem xp1d2m1eqxm1d2 8979
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 7904 . . . 4  |-  ( X  e.  CC  ->  ( X  +  1 )  e.  CC )
21halfcld 8971 . . 3  |-  ( X  e.  CC  ->  (
( X  +  1 )  /  2 )  e.  CC )
3 peano2cnm 8035 . . 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 8035 . . 3  |-  ( X  e.  CC  ->  ( X  -  1 )  e.  CC )
65halfcld 8971 . 2  |-  ( X  e.  CC  ->  (
( X  -  1 )  /  2 )  e.  CC )
7 2cnd 8800 . 2  |-  ( X  e.  CC  ->  2  e.  CC )
8 2ap0 8820 . . 3  |-  2 #  0
98a1i 9 . 2  |-  ( X  e.  CC  ->  2 #  0 )
10 1cnd 7789 . . . 4  |-  ( X  e.  CC  ->  1  e.  CC )
112, 10, 7subdird 8184 . . 3  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  -  1 )  x.  2 )  =  ( ( ( ( X  +  1 )  /  2 )  x.  2 )  -  ( 1  x.  2 ) ) )
121, 7, 9divcanap1d 8558 . . . 4  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  x.  2 )  =  ( X  + 
1 ) )
137mulid2d 7791 . . . 4  |-  ( X  e.  CC  ->  (
1  x.  2 )  =  2 )
1412, 13oveq12d 5792 . . 3  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( X  +  1 )  - 
2 ) )
155, 7, 9divcanap1d 8558 . . . 4  |-  ( X  e.  CC  ->  (
( ( X  - 
1 )  /  2
)  x.  2 )  =  ( X  - 
1 ) )
16 2m1e1 8845 . . . . . 6  |-  ( 2  -  1 )  =  1
1716a1i 9 . . . . 5  |-  ( X  e.  CC  ->  (
2  -  1 )  =  1 )
1817oveq2d 5790 . . . 4  |-  ( X  e.  CC  ->  ( X  -  ( 2  -  1 ) )  =  ( X  - 
1 ) )
19 id 19 . . . . 5  |-  ( X  e.  CC  ->  X  e.  CC )
2019, 7, 10subsub3d 8110 . . . 4  |-  ( X  e.  CC  ->  ( X  -  ( 2  -  1 ) )  =  ( ( X  +  1 )  - 
2 ) )
2115, 18, 203eqtr2rd 2179 . . 3  |-  ( X  e.  CC  ->  (
( X  +  1 )  -  2 )  =  ( ( ( X  -  1 )  /  2 )  x.  2 ) )
2211, 14, 213eqtrd 2176 . 2  |-  ( X  e.  CC  ->  (
( ( ( X  +  1 )  / 
2 )  -  1 )  x.  2 )  =  ( ( ( X  -  1 )  /  2 )  x.  2 ) )
234, 6, 7, 9, 22mulcanap2ad 8432 1  |-  ( X  e.  CC  ->  (
( ( X  + 
1 )  /  2
)  -  1 )  =  ( ( X  -  1 )  / 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632    - cmin 7940   # cap 8350    / cdiv 8439   2c2 8778
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-2 8786
This theorem is referenced by:  zob  11595  nno  11610  nn0ob  11612
  Copyright terms: Public domain W3C validator