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Theorem eluzgtdifelfzo 10226
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 109 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  N  e.  ( ZZ>= `  A )
)
21adantl 277 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  ( ZZ>= `  A )
)
3 simpl 109 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
43adantr 276 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  A  e.  ZZ )
5 eluzelz 9566 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  ZZ )
65ad2antrr 488 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  N  e.  ZZ )
7 simprr 531 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  B  e.  ZZ )
86, 7zsubcld 9409 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  ( N  -  B )  e.  ZZ )
98ancoms 268 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  -  B )  e.  ZZ )
104, 9zaddcld 9408 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( A  +  ( N  -  B ) )  e.  ZZ )
11 zre 9286 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  RR )
12 zre 9286 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
13 posdif 8441 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  0  <  ( A  -  B ) ) )
1413biimpd 144 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  ->  0  <  ( A  -  B ) ) )
1511, 12, 14syl2anr 290 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  <  A  ->  0  <  ( A  -  B ) ) )
1615adantld 278 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  0  <  ( A  -  B )
) )
1716imp 124 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  0  <  ( A  -  B ) )
18 resubcl 8250 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
1912, 11, 18syl2an 289 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  RR )
2019adantr 276 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( A  -  B )  e.  RR )
21 eluzelre 9567 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  RR )
2221ad2antrl 490 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  RR )
2320, 22ltaddposd 8515 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( 0  <  ( A  -  B )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
2417, 23mpbid 147 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  <  ( N  +  ( A  -  B ) ) )
25 zcn 9287 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
2625ad2antrr 488 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  A  e.  CC )
27 eluzelcn 9568 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  CC )
2827ad2antrl 490 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  CC )
29 zcn 9287 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
3029adantl 277 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
3130adantr 276 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  B  e.  CC )
32 addsub12 8199 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( N  -  B ) )  =  ( N  +  ( A  -  B ) ) )
3332breq2d 4030 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  B  e.  CC )  ->  ( N  <  ( A  +  ( N  -  B
) )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
3426, 28, 31, 33syl3anc 1249 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  <  ( A  +  ( N  -  B ) )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
3524, 34mpbird 167 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  <  ( A  +  ( N  -  B ) ) )
36 elfzo2 10179 . . . 4  |-  ( N  e.  ( A..^ ( A  +  ( N  -  B ) ) )  <-> 
( N  e.  (
ZZ>= `  A )  /\  ( A  +  ( N  -  B )
)  e.  ZZ  /\  N  <  ( A  +  ( N  -  B
) ) ) )
372, 10, 35, 36syl3anbrc 1183 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  ( A..^ ( A  +  ( N  -  B
) ) ) )
38 fzosubel3 10225 . . 3  |-  ( ( N  e.  ( A..^ ( A  +  ( N  -  B ) ) )  /\  ( N  -  B )  e.  ZZ )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) )
3937, 9, 38syl2anc 411 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) )
4039ex 115 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 104    <-> wb 105    /\ w3a 980    e. wcel 2160   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840    + caddc 7843    < clt 8021    - cmin 8157   ZZcz 9282   ZZ>=cuz 9557  ..^cfzo 10171
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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-ltadd 7956
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-inn 8949  df-n0 9206  df-z 9283  df-uz 9558  df-fz 10038  df-fzo 10172
This theorem is referenced by:  ige2m2fzo  10227
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