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Theorem fz0fzelfz0 10083
Description: If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
Assertion
Ref Expression
fz0fzelfz0 ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))

Proof of Theorem fz0fzelfz0
StepHypRef Expression
1 elfz2nn0 10068 . . . 4 (𝑁 ∈ (0...𝑅) ↔ (𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅))
2 elfz2 9972 . . . . . 6 (𝑀 ∈ (𝑁...𝑅) ↔ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)))
3 simplr 525 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑀 ∈ ℤ) ∧ 𝑁𝑀) → 𝑀 ∈ ℤ)
4 0red 7921 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝑀 ∈ ℤ) → 0 ∈ ℝ)
5 nn0re 9144 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
65adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝑀 ∈ ℤ) → 𝑁 ∈ ℝ)
7 zre 9216 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
87adantl 275 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝑀 ∈ ℤ) → 𝑀 ∈ ℝ)
94, 6, 83jca 1172 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0𝑀 ∈ ℤ) → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ))
109adantr 274 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑀 ∈ ℤ) ∧ 𝑁𝑀) → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ))
11 nn0ge0 9160 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
1211adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0𝑀 ∈ ℤ) → 0 ≤ 𝑁)
1312anim1i 338 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑀 ∈ ℤ) ∧ 𝑁𝑀) → (0 ≤ 𝑁𝑁𝑀))
14 letr 8002 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁𝑁𝑀) → 0 ≤ 𝑀))
1510, 13, 14sylc 62 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑀 ∈ ℤ) ∧ 𝑁𝑀) → 0 ≤ 𝑀)
16 elnn0z 9225 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀))
173, 15, 16sylanbrc 415 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑀 ∈ ℤ) ∧ 𝑁𝑀) → 𝑀 ∈ ℕ0)
1817exp31 362 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℤ → (𝑁𝑀𝑀 ∈ ℕ0)))
1918com23 78 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁𝑀 → (𝑀 ∈ ℤ → 𝑀 ∈ ℕ0)))
20193ad2ant1 1013 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → (𝑁𝑀 → (𝑀 ∈ ℤ → 𝑀 ∈ ℕ0)))
2120com13 80 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → (𝑁𝑀 → ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → 𝑀 ∈ ℕ0)))
2221adantrd 277 . . . . . . . . . . 11 (𝑀 ∈ ℤ → ((𝑁𝑀𝑀𝑅) → ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → 𝑀 ∈ ℕ0)))
23223ad2ant3 1015 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁𝑀𝑀𝑅) → ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → 𝑀 ∈ ℕ0)))
2423imp 123 . . . . . . . . 9 (((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)) → ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → 𝑀 ∈ ℕ0))
2524imp 123 . . . . . . . 8 ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)) ∧ (𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅)) → 𝑀 ∈ ℕ0)
26 simpr2 999 . . . . . . . 8 ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)) ∧ (𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅)) → 𝑅 ∈ ℕ0)
27 simplrr 531 . . . . . . . 8 ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)) ∧ (𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅)) → 𝑀𝑅)
2825, 26, 273jca 1172 . . . . . . 7 ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)) ∧ (𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅)) → (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅))
2928ex 114 . . . . . 6 (((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁𝑀𝑀𝑅)) → ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅)))
302, 29sylbi 120 . . . . 5 (𝑀 ∈ (𝑁...𝑅) → ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅)))
3130com12 30 . . . 4 ((𝑁 ∈ ℕ0𝑅 ∈ ℕ0𝑁𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅)))
321, 31sylbi 120 . . 3 (𝑁 ∈ (0...𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅)))
3332imp 123 . 2 ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅))
34 elfz2nn0 10068 . 2 (𝑀 ∈ (0...𝑅) ↔ (𝑀 ∈ ℕ0𝑅 ∈ ℕ0𝑀𝑅))
3533, 34sylibr 133 1 ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973  wcel 2141   class class class wbr 3989  (class class class)co 5853  cr 7773  0cc0 7774  cle 7955  0cn0 9135  cz 9212  ...cfz 9965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966
This theorem is referenced by:  fz0fzdiffz0  10086
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