ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fz0fzdiffz0 GIF version

Theorem fz0fzdiffz0 10132
Description: The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
Assertion
Ref Expression
fz0fzdiffz0 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾𝑀) ∈ (0...𝑁))

Proof of Theorem fz0fzdiffz0
StepHypRef Expression
1 fz0fzelfz0 10129 . . 3 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (0...𝑁))
2 elfzle1 10029 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑀𝐾)
32adantl 277 . . . . . 6 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑀𝐾)
43adantl 277 . . . . 5 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → 𝑀𝐾)
5 elfznn0 10116 . . . . . . 7 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
65adantr 276 . . . . . 6 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℕ0)
7 elfznn0 10116 . . . . . 6 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
8 nn0sub 9321 . . . . . 6 ((𝑀 ∈ ℕ0𝐾 ∈ ℕ0) → (𝑀𝐾 ↔ (𝐾𝑀) ∈ ℕ0))
96, 7, 8syl2anr 290 . . . . 5 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → (𝑀𝐾 ↔ (𝐾𝑀) ∈ ℕ0))
104, 9mpbid 147 . . . 4 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → (𝐾𝑀) ∈ ℕ0)
11 elfz3nn0 10117 . . . . 5 (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
1211adantr 276 . . . 4 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → 𝑁 ∈ ℕ0)
13 elfz2nn0 10114 . . . . . . 7 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
14 elfz2 10017 . . . . . . . . . . 11 (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀𝐾𝐾𝑁)))
15 zsubcl 9296 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾𝑀) ∈ ℤ)
1615zred 9377 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾𝑀) ∈ ℝ)
1716ancoms 268 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾𝑀) ∈ ℝ)
18173adant2 1016 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾𝑀) ∈ ℝ)
19 zre 9259 . . . . . . . . . . . . . . . . . . . . 21 (𝐾 ∈ ℤ → 𝐾 ∈ ℝ)
20193ad2ant3 1020 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℝ)
21 zre 9259 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
22213ad2ant2 1019 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℝ)
2318, 20, 223jca 1177 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ))
2423adantr 276 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → ((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ))
2524adantr 276 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) ∧ 𝐾𝑁) → ((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ))
26 nn0ge0 9203 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0 → 0 ≤ 𝑀)
2726adantl 277 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → 0 ≤ 𝑀)
28 nn0re 9187 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
29 subge02 8437 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ≤ 𝑀 ↔ (𝐾𝑀) ≤ 𝐾))
3020, 28, 29syl2an 289 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → (0 ≤ 𝑀 ↔ (𝐾𝑀) ≤ 𝐾))
3127, 30mpbid 147 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → (𝐾𝑀) ≤ 𝐾)
3231anim1i 340 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) ∧ 𝐾𝑁) → ((𝐾𝑀) ≤ 𝐾𝐾𝑁))
33 letr 8042 . . . . . . . . . . . . . . . . 17 (((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝐾𝑀) ≤ 𝐾𝐾𝑁) → (𝐾𝑀) ≤ 𝑁))
3425, 32, 33sylc 62 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) ∧ 𝐾𝑁) → (𝐾𝑀) ≤ 𝑁)
3534exp31 364 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℕ0 → (𝐾𝑁 → (𝐾𝑀) ≤ 𝑁)))
3635a1i 9 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℕ0 → (𝐾𝑁 → (𝐾𝑀) ≤ 𝑁))))
3736com14 88 . . . . . . . . . . . . 13 (𝐾𝑁 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁))))
3837adantl 277 . . . . . . . . . . . 12 ((𝑀𝐾𝐾𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁))))
3938impcom 125 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀𝐾𝐾𝑁)) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁)))
4014, 39sylbi 121 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁)))
4140com13 80 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁)))
4241impcom 125 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁))
43423adant3 1017 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁))
4413, 43sylbi 121 . . . . . 6 (𝑀 ∈ (0...𝑁) → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁))
4544imp 124 . . . . 5 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾𝑀) ≤ 𝑁)
4645adantl 277 . . . 4 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → (𝐾𝑀) ≤ 𝑁)
4710, 12, 463jca 1177 . . 3 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → ((𝐾𝑀) ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾𝑀) ≤ 𝑁))
481, 47mpancom 422 . 2 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐾𝑀) ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾𝑀) ≤ 𝑁))
49 elfz2nn0 10114 . 2 ((𝐾𝑀) ∈ (0...𝑁) ↔ ((𝐾𝑀) ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾𝑀) ≤ 𝑁))
5048, 49sylibr 134 1 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾𝑀) ∈ (0...𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978  wcel 2148   class class class wbr 4005  (class class class)co 5877  cr 7812  0cc0 7813  cle 7995  cmin 8130  0cn0 9178  cz 9255  ...cfz 10010
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator