fzval2 - Metamath Proof Explorer

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Theorem fzval2 13519

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 13518	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
2		zssre 12595	. . . . . . 7 ⊢ ℤ ⊆ ℝ
3		ressxr 11288	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
4	2, 3	sstri 3989	. . . . . 6 ⊢ ℤ ⊆ ℝ^*
5	4	sseli 3976	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ^*)
6	4	sseli 3976	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ^*)
7		iccval 13395	. . . . 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 4211	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ))
10		inrab2 4308	. . . 4 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
11		sseqin2 4215	. . . . . 6 ⊢ (ℤ ⊆ ℝ^* ↔ (ℝ^* ∩ ℤ) = ℤ)
12	4, 11	mpbi 229	. . . . 5 ⊢ (ℝ^* ∩ ℤ) = ℤ
13	12	rabeqi 3442	. . . 4 ⊢ {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
14	10, 13	eqtri 2756	. . 3 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
15	9, 14	eqtr2di 2785	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
16	1, 15	eqtrd 2768	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5148 (class class class)co 7420 ℝcr 11137 ℝ^*cxr 11277 ≤ cle 11279 ℤcz 12588 [,]cicc 13359 ...cfz 13516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-xr 11282 df-neg 11477 df-z 12589 df-icc 13363 df-fz 13517

This theorem is referenced by: dvfsumle 25953 dvfsumleOLD 25954 dvfsumabs 25956 taylplem1 26296 taylplem2 26297 taylpfval 26298 dvtaylp 26304 ppisval 27035

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