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

Theorem fzmmmeqm 10124
Description: Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
Assertion
Ref Expression
fzmmmeqm  |-  ( M  e.  ( L ... N )  ->  (
( N  -  L
)  -  ( M  -  L ) )  =  ( N  -  M ) )

Proof of Theorem fzmmmeqm
StepHypRef Expression
1 elfz2 10081 . . 3  |-  ( M  e.  ( L ... N )  <->  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  /\  ( L  <_  M  /\  M  <_  N ) ) )
2 zcn 9322 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 zcn 9322 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
4 zcn 9322 . . . . . 6  |-  ( L  e.  ZZ  ->  L  e.  CC )
52, 3, 43anim123i 1186 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  L  e.  ZZ )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
653comr 1213 . . . 4  |-  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
76adantr 276 . . 3  |-  ( ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  /\  ( L  <_  M  /\  M  <_  N ) )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
81, 7sylbi 121 . 2  |-  ( M  e.  ( L ... N )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
9 nnncan2 8256 . 2  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC )  ->  (
( N  -  L
)  -  ( M  -  L ) )  =  ( N  -  M ) )
108, 9syl 14 1  |-  ( M  e.  ( L ... N )  ->  (
( N  -  L
)  -  ( M  -  L ) )  =  ( N  -  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870    <_ cle 8055    - cmin 8190   ZZcz 9317   ...cfz 10074
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sub 8192  df-neg 8193  df-z 9318  df-fz 10075
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator