| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 13549 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) | |
| 2 | zssre 12620 | . . . . . . 7 ⊢ ℤ ⊆ ℝ | |
| 3 | ressxr 11305 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* | |
| 4 | 2, 3 | sstri 3993 | . . . . . 6 ⊢ ℤ ⊆ ℝ* |
| 5 | 4 | sseli 3979 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*) |
| 6 | 4 | sseli 3979 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ*) |
| 7 | iccval 13426 | . . . . 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 4219 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ)) |
| 10 | inrab2 4317 | . . . 4 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} | |
| 11 | sseqin2 4223 | . . . . . 6 ⊢ (ℤ ⊆ ℝ* ↔ (ℝ* ∩ ℤ) = ℤ) | |
| 12 | 4, 11 | mpbi 230 | . . . . 5 ⊢ (ℝ* ∩ ℤ) = ℤ |
| 13 | 12 | rabeqi 3450 | . . . 4 ⊢ {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} |
| 14 | 10, 13 | eqtri 2765 | . . 3 ⊢ ({𝑘 ∈ ℝ* ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} |
| 15 | 9, 14 | eqtr2di 2794 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} = ((𝑀[,]𝑁) ∩ ℤ)) |
| 16 | 1, 15 | eqtrd 2777 | 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 2108 {crab 3436 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 ℝ*cxr 11294 ≤ cle 11296 ℤcz 12613 [,]cicc 13390 ...cfz 13547 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-neg 11495 df-z 12614 df-icc 13394 df-fz 13548 |
| This theorem is referenced by: dvfsumle 26060 dvfsumleOLD 26061 dvfsumabs 26063 taylplem1 26404 taylplem2 26405 taylpfval 26406 dvtaylp 26412 ppisval 27147 |
| Copyright terms: Public domain | W3C validator |