Theorem fz0fzelfz0 12445

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 12431	. . . 4 ⊢ (𝑁 ∈ (0...𝑅) ↔ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅))
2		elfz2 12333	. . . . . 6 ⊢ (𝑀 ∈ (𝑁...𝑅) ↔ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)))
3		simplr 792	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 𝑀 ∈ ℤ)
4		0red 10041	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 0 ∈ ℝ)
5		nn0re 11301	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
6	5	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 𝑁 ∈ ℝ)
7		zre 11381	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
8	7	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ)
9	4, 6, 8	3jca 1242	. . . . . . . . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12	11	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 0 ≤ 𝑁)
13	12	anim1i 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → (0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))
14		letr 10131	. . . . . . . . . . . . . . . . . 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 11390	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀))
17	3, 15, 16	sylanbrc 698	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 𝑀 ∈ ℕ0)
18	17	exp31 630	. . . . . . . . . . . . . . 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 1082	. . . . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
23	22	3ad2ant3 1084	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
24	23	imp 445	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0))
25	24	imp 445	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑀 ∈ ℕ0)
26		simpr2 1068	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑅 ∈ ℕ0)
27		simplrr 801	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑀 ≤ 𝑅)
28	25, 26, 27	3jca 1242	. . . . . . 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 450	. . . . . 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 207	. . . . 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 207	. . 3 ⊢ (𝑁 ∈ (0...𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
33	32	imp 445	. 2 ⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
34		elfz2nn0 12431	. 2 ⊢ (𝑀 ∈ (0...𝑅) ↔ (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
35	33, 34	sylibr 224	1 ⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   class class class wbr 4653  (class class class)co 6650  ℝcr 9935  0cc0 9936   ≤ cle 10075  ℕ₀cn0 11292  ℤcz 11377  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  fz0fzdiffz0  12448  fourierdlem15  40339