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Theorem fz0fzdiffz0 9800
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 9797 . . 3 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (0...𝑁))
2 elfzle1 9700 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑀𝐾)
32adantl 273 . . . . . 6 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑀𝐾)
43adantl 273 . . . . 5 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → 𝑀𝐾)
5 elfznn0 9787 . . . . . . 7 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
65adantr 272 . . . . . 6 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℕ0)
7 elfznn0 9787 . . . . . 6 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
8 nn0sub 9024 . . . . . 6 ((𝑀 ∈ ℕ0𝐾 ∈ ℕ0) → (𝑀𝐾 ↔ (𝐾𝑀) ∈ ℕ0))
96, 7, 8syl2anr 286 . . . . 5 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → (𝑀𝐾 ↔ (𝐾𝑀) ∈ ℕ0))
104, 9mpbid 146 . . . 4 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → (𝐾𝑀) ∈ ℕ0)
11 elfz3nn0 9788 . . . . 5 (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
1211adantr 272 . . . 4 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → 𝑁 ∈ ℕ0)
13 elfz2nn0 9785 . . . . . . 7 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
14 elfz2 9690 . . . . . . . . . . 11 (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀𝐾𝐾𝑁)))
15 zsubcl 8999 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾𝑀) ∈ ℤ)
1615zred 9077 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾𝑀) ∈ ℝ)
1716ancoms 266 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾𝑀) ∈ ℝ)
18173adant2 983 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾𝑀) ∈ ℝ)
19 zre 8962 . . . . . . . . . . . . . . . . . . . . 21 (𝐾 ∈ ℤ → 𝐾 ∈ ℝ)
20193ad2ant3 987 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℝ)
21 zre 8962 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
22213ad2ant2 986 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℝ)
2318, 20, 223jca 1144 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ))
2423adantr 272 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → ((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ))
2524adantr 272 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) ∧ 𝐾𝑁) → ((𝐾𝑀) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ))
26 nn0ge0 8906 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0 → 0 ≤ 𝑀)
2726adantl 273 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → 0 ≤ 𝑀)
28 nn0re 8890 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
29 subge02 8159 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ≤ 𝑀 ↔ (𝐾𝑀) ≤ 𝐾))
3020, 28, 29syl2an 285 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → (0 ≤ 𝑀 ↔ (𝐾𝑀) ≤ 𝐾))
3127, 30mpbid 146 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) → (𝐾𝑀) ≤ 𝐾)
3231anim1i 336 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℕ0) ∧ 𝐾𝑁) → ((𝐾𝑀) ≤ 𝐾𝐾𝑁))
33 letr 7770 . . . . . . . . . . . . . . . . 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 359 . . . . . . . . . . . . . . 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 273 . . . . . . . . . . . 12 ((𝑀𝐾𝐾𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁))))
3938impcom 124 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀𝐾𝐾𝑁)) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁)))
4014, 39sylbi 120 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝐾𝑀) ≤ 𝑁)))
4140com13 80 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁)))
4241impcom 124 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁))
43423adant3 984 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁))
4413, 43sylbi 120 . . . . . 6 (𝑀 ∈ (0...𝑁) → (𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ≤ 𝑁))
4544imp 123 . . . . 5 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾𝑀) ≤ 𝑁)
4645adantl 273 . . . 4 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → (𝐾𝑀) ≤ 𝑁)
4710, 12, 463jca 1144 . . 3 ((𝐾 ∈ (0...𝑁) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁))) → ((𝐾𝑀) ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾𝑀) ≤ 𝑁))
481, 47mpancom 416 . 2 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐾𝑀) ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾𝑀) ≤ 𝑁))
49 elfz2nn0 9785 . 2 ((𝐾𝑀) ∈ (0...𝑁) ↔ ((𝐾𝑀) ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾𝑀) ≤ 𝑁))
5048, 49sylibr 133 1 ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾𝑀) ∈ (0...𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945  wcel 1463   class class class wbr 3895  (class class class)co 5728  cr 7546  0cc0 7547  cle 7725  cmin 7856  0cn0 8881  cz 8958  ...cfz 9683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-fz 9684
This theorem is referenced by: (None)
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