fzval2 - Metamath Proof Explorer

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Theorem fzval2 13490

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 13489	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
2		zssre 12566	. . . . . . 7 ⊢ ℤ ⊆ ℝ
3		ressxr 11259	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
4	2, 3	sstri 3986	. . . . . 6 ⊢ ℤ ⊆ ℝ^*
5	4	sseli 3973	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ^*)
6	4	sseli 3973	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ^*)
7		iccval 13366	. . . . 5 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ^ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
8	5, 6, 7	syl2an 595	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
9	8	ineq1d 4206	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ))
10		inrab2 4302	. . . 4 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
11		sseqin2 4210	. . . . . 6 ⊢ (ℤ ⊆ ℝ^* ↔ (ℝ^* ∩ ℤ) = ℤ)
12	4, 11	mpbi 229	. . . . 5 ⊢ (ℝ^* ∩ ℤ) = ℤ
13	12	rabeqi 3439	. . . 4 ⊢ {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
14	10, 13	eqtri 2754	. . 3 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
15	9, 14	eqtr2di 2783	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
16	1, 15	eqtrd 2766	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 ∩ cin 3942 ⊆ wss 3943 class class class wbr 5141 (class class class)co 7404 ℝcr 11108 ℝ^*cxr 11248 ≤ cle 11250 ℤcz 12559 [,]cicc 13330 ...cfz 13487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-xr 11253 df-neg 11448 df-z 12560 df-icc 13334 df-fz 13488

This theorem is referenced by: dvfsumle 25905 dvfsumleOLD 25906 dvfsumabs 25908 taylplem1 26248 taylplem2 26249 taylpfval 26250 dvtaylp 26256 ppisval 26987

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