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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 12983 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) | |
2 | zssre 12069 | . . . . . . 7 ⊢ ℤ ⊆ ℝ | |
3 | ressxr 10763 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* | |
4 | 2, 3 | sstri 3886 | . . . . . 6 ⊢ ℤ ⊆ ℝ* |
5 | 4 | sseli 3873 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*) |
6 | 4 | sseli 3873 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ*) |
7 | iccval 12860 | . . . . 5 ⊢ ((𝑀 ∈ ℝ* ∧ 𝑁 ∈ ℝ*) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) | |
8 | 5, 6, 7 | syl2an 599 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
9 | 8 | ineq1d 4102 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ)) |
10 | inrab2 4196 | . . . 4 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} | |
11 | sseqin2 4106 | . . . . . 6 ⊢ (ℤ ⊆ ℝ* ↔ (ℝ* ∩ ℤ) = ℤ) | |
12 | 4, 11 | mpbi 233 | . . . . 5 ⊢ (ℝ* ∩ ℤ) = ℤ |
13 | 12 | rabeqi 3383 | . . . 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 399 = wceq 1542 ∈ wcel 2114 {crab 3057 ∩ cin 3842 ⊆ wss 3843 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 ℝ*cxr 10752 ≤ cle 10754 ℤcz 12062 [,]cicc 12824 ...cfz 12981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-xr 10757 df-neg 10951 df-z 12063 df-icc 12828 df-fz 12982 |
This theorem is referenced by: dvfsumle 24773 dvfsumabs 24775 taylplem1 25110 taylplem2 25111 taylpfval 25112 dvtaylp 25117 ppisval 25841 |
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