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Theorem ubmelm1fzo 10135
Description: The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)
Assertion
Ref Expression
ubmelm1fzo  |-  ( K  e.  ( 0..^ N )  ->  ( ( N  -  K )  -  1 )  e.  ( 0..^ N ) )

Proof of Theorem ubmelm1fzo
StepHypRef Expression
1 elfzo0 10091 . 2  |-  ( K  e.  ( 0..^ N )  <->  ( K  e. 
NN0  /\  N  e.  NN  /\  K  <  N
) )
2 nnz 9192 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
32adantr 274 . . . . . . . 8  |-  ( ( N  e.  NN  /\  K  e.  NN0 )  ->  N  e.  ZZ )
4 nn0z 9193 . . . . . . . . 9  |-  ( K  e.  NN0  ->  K  e.  ZZ )
54adantl 275 . . . . . . . 8  |-  ( ( N  e.  NN  /\  K  e.  NN0 )  ->  K  e.  ZZ )
63, 5zsubcld 9297 . . . . . . 7  |-  ( ( N  e.  NN  /\  K  e.  NN0 )  -> 
( N  -  K
)  e.  ZZ )
76ancoms 266 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( N  -  K
)  e.  ZZ )
8 peano2zm 9211 . . . . . 6  |-  ( ( N  -  K )  e.  ZZ  ->  (
( N  -  K
)  -  1 )  e.  ZZ )
97, 8syl 14 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( ( N  -  K )  -  1 )  e.  ZZ )
1093adant3 1002 . . . 4  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  (
( N  -  K
)  -  1 )  e.  ZZ )
11 simp3 984 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  K  <  N )
124, 2anim12i 336 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
13123adant3 1002 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
14 znnsub 9224 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  N  <->  ( N  -  K )  e.  NN ) )
1513, 14syl 14 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( K  <  N  <->  ( N  -  K )  e.  NN ) )
1611, 15mpbid 146 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( N  -  K )  e.  NN )
17 nnm1ge0 9256 . . . . 5  |-  ( ( N  -  K )  e.  NN  ->  0  <_  ( ( N  -  K )  -  1 ) )
1816, 17syl 14 . . . 4  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  0  <_  ( ( N  -  K )  -  1 ) )
19 elnn0z 9186 . . . 4  |-  ( ( ( N  -  K
)  -  1 )  e.  NN0  <->  ( ( ( N  -  K )  -  1 )  e.  ZZ  /\  0  <_ 
( ( N  -  K )  -  1 ) ) )
2010, 18, 19sylanbrc 414 . . 3  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  (
( N  -  K
)  -  1 )  e.  NN0 )
21 simp2 983 . . 3  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  N  e.  NN )
22 nncn 8847 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
2322adantl 275 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  N  e.  CC )
24 nn0cn 9106 . . . . . . 7  |-  ( K  e.  NN0  ->  K  e.  CC )
2524adantr 274 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  K  e.  CC )
26 1cnd 7897 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  1  e.  CC )
2723, 25, 26subsub4d 8222 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( ( N  -  K )  -  1 )  =  ( N  -  ( K  + 
1 ) ) )
28 nn0p1gt0 9125 . . . . . . 7  |-  ( K  e.  NN0  ->  0  < 
( K  +  1 ) )
2928adantr 274 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  0  <  ( K  +  1 ) )
30 nn0re 9105 . . . . . . . 8  |-  ( K  e.  NN0  ->  K  e.  RR )
31 peano2re 8016 . . . . . . . 8  |-  ( K  e.  RR  ->  ( K  +  1 )  e.  RR )
3230, 31syl 14 . . . . . . 7  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  RR )
33 nnre 8846 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR )
34 ltsubpos 8334 . . . . . . 7  |-  ( ( ( K  +  1 )  e.  RR  /\  N  e.  RR )  ->  ( 0  <  ( K  +  1 )  <-> 
( N  -  ( K  +  1 ) )  <  N ) )
3532, 33, 34syl2an 287 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( 0  <  ( K  +  1 )  <-> 
( N  -  ( K  +  1 ) )  <  N ) )
3629, 35mpbid 146 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( N  -  ( K  +  1 ) )  <  N )
3727, 36eqbrtrd 3989 . . . 4  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( ( N  -  K )  -  1 )  <  N )
38373adant3 1002 . . 3  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  (
( N  -  K
)  -  1 )  <  N )
39 elfzo0 10091 . . 3  |-  ( ( ( N  -  K
)  -  1 )  e.  ( 0..^ N )  <->  ( ( ( N  -  K )  -  1 )  e. 
NN0  /\  N  e.  NN  /\  ( ( N  -  K )  - 
1 )  <  N
) )
4020, 21, 38, 39syl3anbrc 1166 . 2  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  (
( N  -  K
)  -  1 )  e.  ( 0..^ N ) )
411, 40sylbi 120 1  |-  ( K  e.  ( 0..^ N )  ->  ( ( N  -  K )  -  1 )  e.  ( 0..^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    e. wcel 2128   class class class wbr 3967  (class class class)co 5827   CCcc 7733   RRcr 7734   0cc0 7735   1c1 7736    + caddc 7738    < clt 7915    <_ cle 7916    - cmin 8051   NNcn 8839   NN0cn0 9096   ZZcz 9173  ..^cfzo 10051
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-addcom 7835  ax-addass 7837  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-0id 7843  ax-rnegex 7844  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-ltadd 7851
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-inn 8840  df-n0 9097  df-z 9174  df-uz 9446  df-fz 9920  df-fzo 10052
This theorem is referenced by: (None)
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