fzval2 - Metamath Proof Explorer

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Theorem fzval2 13424

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 13423	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
2		zssre 12493	. . . . . . 7 ⊢ ℤ ⊆ ℝ
3		ressxr 11174	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
4	2, 3	sstri 3941	. . . . . 6 ⊢ ℤ ⊆ ℝ^*
5	4	sseli 3927	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ^*)
6	4	sseli 3927	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ^*)
7		iccval 13298	. . . . 5 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ^ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
8	5, 6, 7	syl2an 596	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
9	8	ineq1d 4169	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ))
10		inrab2 4267	. . . 4 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
11		sseqin2 4173	. . . . . 6 ⊢ (ℤ ⊆ ℝ^* ↔ (ℝ^* ∩ ℤ) = ℤ)
12	4, 11	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ ℤ) = ℤ
13	12	rabeqi 3410	. . . 4 ⊢ {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
14	10, 13	eqtri 2757	. . 3 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
15	9, 14	eqtr2di 2786	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
16	1, 15	eqtrd 2769	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3397 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5096 (class class class)co 7356 ℝcr 11023 ℝ^*cxr 11163 ≤ cle 11165 ℤcz 12486 [,]cicc 13262 ...cfz 13421

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-iota 6446 df-fun 6492 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-xr 11168 df-neg 11365 df-z 12487 df-icc 13266 df-fz 13422

This theorem is referenced by: dvfsumle 25980 dvfsumleOLD 25981 dvfsumabs 25983 taylplem1 26324 taylplem2 26325 taylpfval 26326 dvtaylp 26332 ppisval 27068

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