Theorem fzval2 10219

Description: An alternate 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 10218	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
2		zssre 9464	. . . . . . 7 ⊢ ℤ ⊆ ℝ
3		ressxr 8201	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
4	2, 3	sstri 3233	. . . . . 6 ⊢ ℤ ⊆ ℝ*
5	4	sseli 3220	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
6	4	sseli 3220	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ*)
7		iccval 10128	. . . . 5 ⊢ ((𝑀 ∈ ℝ* ∧ 𝑁 ∈ ℝ*) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
8	5, 6, 7	syl2an 289	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
9	8	ineq1d 3404	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ))
10		inrab2 3477	. . . 4 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
11		sseqin2 3423	. . . . . 6 ⊢ (ℤ ⊆ ℝ* ↔ (ℝ* ∩ ℤ) = ℤ)
12	4, 11	mpbi 145	. . . . 5 ⊢ (ℝ* ∩ ℤ) = ℤ
13		rabeq 2791	. . . . 5 ⊢ ((ℝ* ∩ ℤ) = ℤ → {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
14	12, 13	ax-mp 5	. . . 4 ⊢ {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
15	10, 14	eqtri 2250	. . 3 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
16	9, 15	eqtr2di 2279	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
17	1, 16	eqtrd 2262	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {crab 2512   ∩ cin 3196   ⊆ wss 3197   class class class wbr 4083  (class class class)co 6007  ℝcr 8009  ℝ^*cxr 8191   ≤ cle 8193  ℤcz 9457  [,]cicc 10099  ...cfz 10216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-neg 8331  df-z 9458  df-icc 10103  df-fz 10217

This theorem is referenced by: (None)