fzval2 - Metamath Proof Explorer

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Theorem fzval2 13447

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 13446	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
2		zssre 12512	. . . . . . 7 ⊢ ℤ ⊆ ℝ
3		ressxr 11194	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
4	2, 3	sstri 3953	. . . . . 6 ⊢ ℤ ⊆ ℝ^*
5	4	sseli 3939	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ^*)
6	4	sseli 3939	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ^*)
7		iccval 13321	. . . . 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 4178	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ))
10		inrab2 4276	. . . 4 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
11		sseqin2 4182	. . . . . 6 ⊢ (ℤ ⊆ ℝ^* ↔ (ℝ^* ∩ ℤ) = ℤ)
12	4, 11	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ ℤ) = ℤ
13	12	rabeqi 3416	. . . 4 ⊢ {𝑘 ∈ (ℝ^* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
14	10, 13	eqtri 2752	. . 3 ⊢ ({𝑘 ∈ ℝ^* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}
15	9, 14	eqtr2di 2781	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
16	1, 15	eqtrd 2764	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3402 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5102 (class class class)co 7369 ℝcr 11043 ℝ^*cxr 11183 ≤ cle 11185 ℤcz 12505 [,]cicc 13285 ...cfz 13444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6452 df-fun 6501 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-xr 11188 df-neg 11384 df-z 12506 df-icc 13289 df-fz 13445

This theorem is referenced by: dvfsumle 25959 dvfsumleOLD 25960 dvfsumabs 25962 taylplem1 26303 taylplem2 26304 taylpfval 26305 dvtaylp 26311 ppisval 27047

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