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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 7438 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1)) | |
| 3 | 1, 2 | oveqan12d 7450 | . . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚...(𝑛 − 1)) = (𝑀...(𝑁 − 1))) |
| 4 | df-fzo 13695 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
| 5 | ovex 7464 | . . 3 ⊢ (𝑀...(𝑁 − 1)) ∈ V | |
| 6 | 3, 4, 5 | ovmpoa 7588 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
| 7 | simpl 482 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ) | |
| 8 | fzof 13696 | . . . . . . 7 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
| 9 | 8 | fdmi 6747 | . . . . . 6 ⊢ dom ..^ = (ℤ × ℤ) |
| 10 | 9 | ndmov 7617 | . . . . 5 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = ∅) |
| 11 | 7, 10 | nsyl5 159 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (𝑀..^𝑁) = ∅) |
| 12 | simpl 482 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → 𝑀 ∈ ℤ) | |
| 13 | fzf 13551 | . . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
| 14 | 13 | fdmi 6747 | . . . . . 6 ⊢ dom ... = (ℤ × ℤ) |
| 15 | 14 | ndmov 7617 | . . . . 5 ⊢ (¬ (𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀...(𝑁 − 1)) = ∅) |
| 16 | 12, 15 | nsyl5 159 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (𝑀...(𝑁 − 1)) = ∅) |
| 17 | 11, 16 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑀 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
| 18 | 17 | adantr 480 | . 2 ⊢ ((¬ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
| 19 | 6, 18 | pm2.61ian 812 | 1 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 𝒫 cpw 4600 × cxp 5683 (class class class)co 7431 1c1 11156 − cmin 11492 ℤcz 12613 ...cfz 13547 ..^cfzo 13694 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 |
| This theorem is referenced by: elfzo 13701 fzon 13720 fzoss1 13726 fzoss2 13727 elfzolem1 13744 fz1fzo0m1 13750 fzval3 13773 fzo13pr 13788 fzo0to2pr 13789 fzo0to3tp 13791 fzo0to42pr 13792 fzo1to4tp 13793 fzoend 13796 fzofzp1b 13804 elfzom1b 13805 peano2fzor 13813 fzoshftral 13823 zmodfzo 13934 zmodidfzo 13940 fzofi 14015 hashfzo 14468 wrdffz 14573 revcl 14799 revlen 14800 revccat 14804 revrev 14805 revco 14873 fzosump1 15788 telfsumo 15838 fsumparts 15842 geoser 15903 pwdif 15904 pwm1geoser 15905 geo2sum2 15910 dfphi2 16811 reumodprminv 16842 gsumwsubmcl 18850 gsumsgrpccat 18853 gsumwmhm 18858 efgsdmi 19750 efgs1b 19754 efgredlemf 19759 efgredlemd 19762 efgredlemc 19763 efgredlem 19765 cpmadugsumlemF 22882 advlogexp 26697 dchrisumlem1 27533 redwlklem 29689 wlkiswwlks2lem3 29891 wlkiswwlksupgr2 29897 clwlkclwwlklem2a 30017 wlk2v2e 30176 eucrct2eupth 30264 cycpmco2 33153 submat1n 33804 eulerpartlemd 34368 fzssfzo 34554 signstfvn 34584 pthhashvtx 35133 remexz 42105 metakunt20 42225 fzosumm1 42291 bccbc 44364 monoords 45309 stirlinglem12 46100 difltmodne 47344 iccpartiltu 47409 iccpartigtl 47410 iccpartgt 47414 nnsum4primeseven 47787 nnsum4primesevenALTV 47788 nn0sumshdiglemA 48540 nn0sumshdiglemB 48541 |
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