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

Theorem eluzgtdifelfzo 9608
Description: Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
Assertion
Ref Expression
eluzgtdifelfzo  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) ) )

Proof of Theorem eluzgtdifelfzo
StepHypRef Expression
1 simpl 107 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  N  e.  ( ZZ>= `  A )
)
21adantl 271 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  ( ZZ>= `  A )
)
3 simpl 107 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
43adantr 270 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  A  e.  ZZ )
5 eluzelz 9028 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  ZZ )
65ad2antrr 472 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  N  e.  ZZ )
7 simprr 499 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  B  e.  ZZ )
86, 7zsubcld 8873 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  ( N  -  B )  e.  ZZ )
98ancoms 264 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  -  B )  e.  ZZ )
104, 9zaddcld 8872 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( A  +  ( N  -  B ) )  e.  ZZ )
11 zre 8754 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  RR )
12 zre 8754 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
13 posdif 7933 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  0  <  ( A  -  B ) ) )
1413biimpd 142 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  ->  0  <  ( A  -  B ) ) )
1511, 12, 14syl2anr 284 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  <  A  ->  0  <  ( A  -  B ) ) )
1615adantld 272 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  0  <  ( A  -  B )
) )
1716imp 122 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  0  <  ( A  -  B ) )
18 resubcl 7746 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
1912, 11, 18syl2an 283 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  RR )
2019adantr 270 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( A  -  B )  e.  RR )
21 eluzelre 9029 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  RR )
2221ad2antrl 474 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  RR )
2320, 22ltaddposd 8006 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( 0  <  ( A  -  B )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
2417, 23mpbid 145 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  <  ( N  +  ( A  -  B ) ) )
25 zcn 8755 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
2625ad2antrr 472 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  A  e.  CC )
27 eluzelcn 9030 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  CC )
2827ad2antrl 474 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  CC )
29 zcn 8755 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
3029adantl 271 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
3130adantr 270 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  B  e.  CC )
32 addsub12 7695 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( N  -  B ) )  =  ( N  +  ( A  -  B ) ) )
3332breq2d 3857 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  B  e.  CC )  ->  ( N  <  ( A  +  ( N  -  B
) )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
3426, 28, 31, 33syl3anc 1174 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  <  ( A  +  ( N  -  B ) )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
3524, 34mpbird 165 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  <  ( A  +  ( N  -  B ) ) )
36 elfzo2 9561 . . . 4  |-  ( N  e.  ( A..^ ( A  +  ( N  -  B ) ) )  <-> 
( N  e.  (
ZZ>= `  A )  /\  ( A  +  ( N  -  B )
)  e.  ZZ  /\  N  <  ( A  +  ( N  -  B
) ) ) )
372, 10, 35, 36syl3anbrc 1127 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  ( A..^ ( A  +  ( N  -  B
) ) ) )
38 fzosubel3 9607 . . 3  |-  ( ( N  e.  ( A..^ ( A  +  ( N  -  B ) ) )  /\  ( N  -  B )  e.  ZZ )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) )
3937, 9, 38syl2anc 403 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) )
4039ex 113 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7348   RRcr 7349   0cc0 7350    + caddc 7353    < clt 7522    - cmin 7653   ZZcz 8750   ZZ>=cuz 9019  ..^cfzo 9553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-fz 9425  df-fzo 9554
This theorem is referenced by:  ige2m2fzo  9609
  Copyright terms: Public domain W3C validator