Theorem fz0fzelfz0 13580

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

Step	Hyp	Ref	Expression
1		elfz2nn0 13564	. . . 4 ⊢ (𝑁 ∈ (0...𝑅) ↔ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅))
2		elfz2 13460	. . . . . 6 ⊢ (𝑀 ∈ (𝑁...𝑅) ↔ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)))
3		simplr 774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 𝑀 ∈ ℤ)
4		0red 11139	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 0 ∈ ℝ)
5		nn0re 12438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
6	5	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 𝑁 ∈ ℝ)
7		zre 12520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
8	7	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ)
9	4, 6, 8	3jca 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ))
10	9	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ))
11		nn0ge0 12454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12	11	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 0 ≤ 𝑁)
13	12	anim1i 621	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → (0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))
14		letr 11232	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 0 ≤ 𝑀))
15	10, 13, 14	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 0 ≤ 𝑀)
16		elnn0z 12529	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀))
17	3, 15, 16	sylanbrc 589	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 𝑀 ∈ ℕ0)
18	17	exp31 420	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑀 ∈ ℤ → (𝑁 ≤ 𝑀 → 𝑀 ∈ ℕ0)))
19	18	com23 86	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 𝑀 → (𝑀 ∈ ℤ → 𝑀 ∈ ℕ0)))
20	19	3ad2ant1 1139	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑁 ≤ 𝑀 → (𝑀 ∈ ℤ → 𝑀 ∈ ℕ0)))
21	20	com13 88	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝑁 ≤ 𝑀 → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
22	21	adantrd 492	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
23	22	3ad2ant3 1141	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
24	23	imp 407	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0))
25	24	imp 407	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑀 ∈ ℕ0)
26		simpr2 1202	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑅 ∈ ℕ0)
27		simplrr 783	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑀 ≤ 𝑅)
28	25, 26, 27	3jca 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
29	28	ex 413	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
30	2, 29	sylbi 218	. . . . 5 ⊢ (𝑀 ∈ (𝑁...𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
31	30	com12 32	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
32	1, 31	sylbi 218	. . 3 ⊢ (𝑁 ∈ (0...𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
33	32	imp 407	. 2 ⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
34		elfz2nn0 13564	. 2 ⊢ (𝑀 ∈ (0...𝑅) ↔ (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
35	33, 34	sylibr 235	1 ⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119   class class class wbr 5073  (class class class)co 7357  ℝcr 11029  0cc0 11030   ≤ cle 11172  ℕ₀cn0 12429  ℤcz 12516  ...cfz 13453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-n0 12430  df-z 12517  df-uz 12781  df-fz 13454

This theorem is referenced by:  fz0fzdiffz0  13583  fourierdlem15  46573