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Mirrors > Home > MPE Home > Th. List > fzoval | Structured version Visualization version GIF version |
Description: Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
fzoval | ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀) | |
2 | oveq1 6556 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1)) | |
3 | 1, 2 | oveqan12d 6568 | . . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚...(𝑛 − 1)) = (𝑀...(𝑁 − 1))) |
4 | df-fzo 12335 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
5 | ovex 6577 | . . 3 ⊢ (𝑀...(𝑁 − 1)) ∈ V | |
6 | 3, 4, 5 | ovmpt2a 6689 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
7 | simpl 472 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ) | |
8 | 7 | con3i 149 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → ¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
9 | fzof 12336 | . . . . . . 7 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
10 | 9 | fdmi 5965 | . . . . . 6 ⊢ dom ..^ = (ℤ × ℤ) |
11 | 10 | ndmov 6716 | . . . . 5 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = ∅) |
12 | 8, 11 | syl 17 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (𝑀..^𝑁) = ∅) |
13 | simpl 472 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → 𝑀 ∈ ℤ) | |
14 | 13 | con3i 149 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → ¬ (𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ)) |
15 | fzf 12201 | . . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
16 | 15 | fdmi 5965 | . . . . . 6 ⊢ dom ... = (ℤ × ℤ) |
17 | 16 | ndmov 6716 | . . . . 5 ⊢ (¬ (𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀...(𝑁 − 1)) = ∅) |
18 | 14, 17 | syl 17 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (𝑀...(𝑁 − 1)) = ∅) |
19 | 12, 18 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑀 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
20 | 19 | adantr 480 | . 2 ⊢ ((¬ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
21 | 6, 20 | pm2.61ian 827 | 1 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 𝒫 cpw 4108 × cxp 5036 (class class class)co 6549 1c1 9816 − cmin 10145 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-neg 10148 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: elfzo 12341 fzon 12358 fzoss1 12364 fzoss2 12365 fz1fzo0m1 12383 fzval3 12404 fzo13pr 12419 fzo0to2pr 12420 fzo0to3tp 12421 fzo0to42pr 12422 fzo1to4tp 12423 fzoend 12425 fzofzp1b 12432 elfzom1b 12433 peano2fzor 12441 fzoshftral 12447 zmodfzo 12555 zmodidfzo 12561 fzofi 12635 hashfzo 13076 wrdffz 13181 revcl 13361 revlen 13362 revccat 13366 revrev 13367 revco 13431 fzosump1 14325 telfsumo 14375 fsumparts 14379 geoser 14438 geo2sum2 14444 dfphi2 15317 reumodprminv 15347 gsumwsubmcl 17198 gsumccat 17201 gsumwmhm 17205 efgsdmi 17968 efgs1b 17972 efgredlemf 17977 efgredlemd 17980 efgredlemc 17981 efgredlem 17983 cpmadugsumlemF 20500 advlogexp 24201 dchrisumlem1 24978 wlkntrllem2 26090 redwlk 26136 constr3pthlem1 26183 constr3pthlem3 26185 wlkiswwlk2lem3 26221 clwlkisclwwlklem2a 26313 submat1n 29199 eulerpartlemd 29755 fzssfzo 29940 signstfvn 29972 bccbc 37566 monoords 38452 elfzolem1 38478 stirlinglem12 38978 iccpartiltu 39960 iccpartigtl 39961 iccpartgt 39965 pwdif 40039 pwm1geoserALT 40040 nnsum4primeseven 40216 nnsum4primesevenALTV 40217 red1wlklem 40880 1wlkiswwlks2lem3 41068 1wlkiswwlksupgr2 41074 clwlkclwwlklem2a 41207 1wlk2v2e 41324 eucrct2eupth 41413 nn0sumshdiglemA 42211 nn0sumshdiglemB 42212 |
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