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Mirrors > Home > MPE Home > Th. List > fzval2 | Structured version Visualization version GIF version |
Description: An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
fzval2 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzval 13432 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) | |
2 | zssre 12511 | . . . . . . 7 ⊢ ℤ ⊆ ℝ | |
3 | ressxr 11204 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* | |
4 | 2, 3 | sstri 3954 | . . . . . 6 ⊢ ℤ ⊆ ℝ* |
5 | 4 | sseli 3941 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*) |
6 | 4 | sseli 3941 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ*) |
7 | iccval 13309 | . . . . 5 ⊢ ((𝑀 ∈ ℝ* ∧ 𝑁 ∈ ℝ*) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) | |
8 | 5, 6, 7 | syl2an 597 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
9 | 8 | ineq1d 4172 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ)) |
10 | inrab2 4268 | . . . 4 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} | |
11 | sseqin2 4176 | . . . . . 6 ⊢ (ℤ ⊆ ℝ* ↔ (ℝ* ∩ ℤ) = ℤ) | |
12 | 4, 11 | mpbi 229 | . . . . 5 ⊢ (ℝ* ∩ ℤ) = ℤ |
13 | 12 | rabeqi 3419 | . . . 4 ⊢ {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} |
14 | 10, 13 | eqtri 2761 | . . 3 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} |
15 | 9, 14 | eqtr2di 2790 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ)) |
16 | 1, 15 | eqtrd 2773 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5106 (class class class)co 7358 ℝcr 11055 ℝ*cxr 11193 ≤ cle 11195 ℤcz 12504 [,]cicc 13273 ...cfz 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-xr 11198 df-neg 11393 df-z 12505 df-icc 13277 df-fz 13431 |
This theorem is referenced by: dvfsumle 25401 dvfsumabs 25403 taylplem1 25738 taylplem2 25739 taylpfval 25740 dvtaylp 25745 ppisval 26469 |
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