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Mirrors > Home > MPE Home > Th. List > fzval | Structured version Visualization version GIF version |
Description: The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where ℕk means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
fzval | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4971 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 ≤ 𝑘 ↔ 𝑀 ≤ 𝑘)) | |
2 | 1 | anbi1d 629 | . . 3 ⊢ (𝑚 = 𝑀 → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛))) |
3 | 2 | rabbidv 3428 | . 2 ⊢ (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) |
4 | breq2 4972 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ 𝑁)) | |
5 | 4 | anbi2d 628 | . . 3 ⊢ (𝑛 = 𝑁 → ((𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁))) |
6 | 5 | rabbidv 3428 | . 2 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
7 | df-fz 12747 | . 2 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) | |
8 | zex 11844 | . . 3 ⊢ ℤ ∈ V | |
9 | 8 | rabex 5133 | . 2 ⊢ {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∈ V |
10 | 3, 6, 7, 9 | ovmpo 7173 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 {crab 3111 class class class wbr 4968 (class class class)co 7023 ≤ cle 10529 ℤcz 11835 ...cfz 12746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-cnex 10446 ax-resscn 10447 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-iota 6196 df-fun 6234 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-neg 10726 df-z 11836 df-fz 12747 |
This theorem is referenced by: fzval2 12749 elfz1 12751 |
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