Theorem fzval2 8877

Description: An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
fzval2	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Proof of Theorem fzval2

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzval 8876	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
2		zssre 8252	. . . . . . 7 ⊢ ℤ ⊆ ℝ
3		ressxr 7069	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
4	2, 3	sstri 2954	. . . . . 6 ⊢ ℤ ⊆ ℝ*
5	4	sseli 2941	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
6	4	sseli 2941	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ*)
7		iccval 8789	. . . . 5 ⊢ ((𝑀 ∈ ℝ* ∧ 𝑁 ∈ ℝ*) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
8	5, 6, 7	syl2an 273	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
9	8	ineq1d 3137	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ))
10		inrab2 3210	. . . 4 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
11		sseqin2 3156	. . . . . 6 ⊢ (ℤ ⊆ ℝ* ↔ (ℝ* ∩ ℤ) = ℤ)
12	4, 11	mpbi 133	. . . . 5 ⊢ (ℝ* ∩ ℤ) = ℤ
13		rabeq 2551	. . . . 5 ⊢ ((ℝ* ∩ ℤ) = ℤ → {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
14	12, 13	ax-mp 7	. . . 4 ⊢ {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
15	10, 14	eqtri 2060	. . 3 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
16	9, 15	syl6req 2089	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
17	1, 16	eqtrd 2072	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {crab 2310   ∩ cin 2916   ⊆ wss 2917   class class class wbr 3764  (class class class)co 5512  ℝcr 6888  ℝ^*cxr 7059   ≤ cle 7061  ℤcz 8245  [,]cicc 8760  ...cfz 8874

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-neg 7185  df-z 8246  df-icc 8764  df-fz 8875

This theorem is referenced by: (None)