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Theorem eluzgtdifelfzo 11881
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 457 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  N  e.  ( ZZ>= `  A )
)
21adantl 466 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  ( ZZ>= `  A )
)
3 simpl 457 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
43adantr 465 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  A  e.  ZZ )
5 eluzelz 11115 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  ZZ )
65ad2antrr 725 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  N  e.  ZZ )
7 simprr 757 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  B  e.  ZZ )
86, 7zsubcld 10995 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  A )  /\  B  <  A )  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  ->  ( N  -  B )  e.  ZZ )
98ancoms 453 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  -  B )  e.  ZZ )
104, 9zaddcld 10994 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( A  +  ( N  -  B ) )  e.  ZZ )
11 zre 10889 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  RR )
12 zre 10889 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
13 posdif 10066 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  0  <  ( A  -  B ) ) )
1413biimpd 207 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  ->  0  <  ( A  -  B ) ) )
1511, 12, 14syl2anr 478 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  <  A  ->  0  <  ( A  -  B ) ) )
1615adantld 467 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  0  <  ( A  -  B )
) )
1716imp 429 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  0  <  ( A  -  B ) )
18 resubcl 9902 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
1912, 11, 18syl2an 477 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  RR )
2019adantr 465 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( A  -  B )  e.  RR )
21 eluzelre 11116 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  RR )
2221ad2antrl 727 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  RR )
2320, 22ltaddposd 10157 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( 0  <  ( A  -  B )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
2417, 23mpbid 210 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  <  ( N  +  ( A  -  B ) ) )
25 zcn 10890 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
2625ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  A  e.  CC )
27 eluzelcn 11117 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  A
)  ->  N  e.  CC )
2827ad2antrl 727 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  CC )
29 zcn 10890 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
3029adantl 466 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
3130adantr 465 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  B  e.  CC )
32 addsub12 9852 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( N  -  B ) )  =  ( N  +  ( A  -  B ) ) )
3332breq2d 4468 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  B  e.  CC )  ->  ( N  <  ( A  +  ( N  -  B
) )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
3426, 28, 31, 33syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  <  ( A  +  ( N  -  B ) )  <->  N  <  ( N  +  ( A  -  B ) ) ) )
3524, 34mpbird 232 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  <  ( A  +  ( N  -  B ) ) )
36 elfzo2 11829 . . . 4  |-  ( N  e.  ( A..^ ( A  +  ( N  -  B ) ) )  <-> 
( N  e.  (
ZZ>= `  A )  /\  ( A  +  ( N  -  B )
)  e.  ZZ  /\  N  <  ( A  +  ( N  -  B
) ) ) )
372, 10, 35, 36syl3anbrc 1180 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  N  e.  ( A..^ ( A  +  ( N  -  B
) ) ) )
38 fzosubel3 11880 . . 3  |-  ( ( N  e.  ( A..^ ( A  +  ( N  -  B ) ) )  /\  ( N  -  B )  e.  ZZ )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) )
3937, 9, 38syl2anc 661 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( N  e.  ( ZZ>= `  A )  /\  B  <  A ) )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) )
4039ex 434 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  ( N  -  A )  e.  ( 0..^ ( N  -  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    < clt 9645    - cmin 9824   ZZcz 10885   ZZ>=cuz 11106  ..^cfzo 11821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822
This theorem is referenced by:  ige2m2fzo  11882
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