Theorem fz0fzelfz0 10545

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 10530	. . . 4 ⊢ (𝑁 ∈ (0...𝑅) ↔ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅))
2		elfz2 10429	. . . . . 6 ⊢ (𝑀 ∈ (𝑁...𝑅) ↔ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)))
3		simplr 533	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 𝑀 ∈ ℤ)
4		0red 8328	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 0 ∈ ℝ)
5		nn0re 9577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
6	5	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 𝑁 ∈ ℝ)
7		zre 9653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
8	7	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ)
9	4, 6, 8	3jca 1208	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ))
10	9	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ))
11		nn0ge0 9593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12	11	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) → 0 ≤ 𝑁)
13	12	anim1i 340	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → (0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))
14		letr 8409	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 0 ≤ 𝑀))
15	10, 13, 14	sylc 62	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 0 ≤ 𝑀)
16		elnn0z 9662	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀))
17	3, 15, 16	sylanbrc 421	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≤ 𝑀) → 𝑀 ∈ ℕ0)
18	17	exp31 364	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑀 ∈ ℤ → (𝑁 ≤ 𝑀 → 𝑀 ∈ ℕ0)))
19	18	com23 78	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 𝑀 → (𝑀 ∈ ℤ → 𝑀 ∈ ℕ0)))
20	19	3ad2ant1 1049	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑁 ≤ 𝑀 → (𝑀 ∈ ℤ → 𝑀 ∈ ℕ0)))
21	20	com13 80	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝑁 ≤ 𝑀 → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
22	21	adantrd 279	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
23	22	3ad2ant3 1051	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0)))
24	23	imp 124	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → 𝑀 ∈ ℕ0))
25	24	imp 124	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑀 ∈ ℕ0)
26		simpr2 1035	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑅 ∈ ℕ0)
27		simplrr 542	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅)) → 𝑀 ≤ 𝑅)
28	25, 26, 27	3jca 1208	. . . . . . 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 115	. . . . . 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 121	. . . . 5 ⊢ (𝑀 ∈ (𝑁...𝑅) → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
31	30	com12 30	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
32	1, 31	sylbi 121	. . 3 ⊢ (𝑁 ∈ (0...𝑅) → (𝑀 ∈ (𝑁...𝑅) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅)))
33	32	imp 124	. 2 ⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
34		elfz2nn0 10530	. 2 ⊢ (𝑀 ∈ (0...𝑅) ↔ (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅))
35	33, 34	sylibr 134	1 ⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   ∈ wcel 2209   class class class wbr 4130  (class class class)co 6085  ℝcr 8179  0cc0 8180   ≤ cle 8362  ℕ₀cn0 9568  ℤcz 9649  ...cfz 10422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932  df-fz 10423

This theorem is used by:  fz0fzdiffz0  10548