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Mirrors > Home > ILE Home > Th. List > fzoval | 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 | elfzoel1 10070 | . . . 4 ⊢ (𝑥 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑥 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)) |
3 | elfzel1 9950 | . . . 4 ⊢ (𝑥 ∈ (𝑀...(𝑁 − 1)) → 𝑀 ∈ ℤ) | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑥 ∈ (𝑀...(𝑁 − 1)) → 𝑀 ∈ ℤ)) |
5 | peano2zm 9220 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
6 | fzf 9939 | . . . . . . . 8 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
7 | 6 | fovcl 5938 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀...(𝑁 − 1)) ∈ 𝒫 ℤ) |
8 | 5, 7 | sylan2 284 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...(𝑁 − 1)) ∈ 𝒫 ℤ) |
9 | id 19 | . . . . . . . 8 ⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) | |
10 | oveq1 5843 | . . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝑧 − 1) = (𝑁 − 1)) | |
11 | 9, 10 | oveqan12d 5855 | . . . . . . 7 ⊢ ((𝑦 = 𝑀 ∧ 𝑧 = 𝑁) → (𝑦...(𝑧 − 1)) = (𝑀...(𝑁 − 1))) |
12 | df-fzo 10068 | . . . . . . 7 ⊢ ..^ = (𝑦 ∈ ℤ, 𝑧 ∈ ℤ ↦ (𝑦...(𝑧 − 1))) | |
13 | 11, 12 | ovmpoga 5962 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀...(𝑁 − 1)) ∈ 𝒫 ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
14 | 8, 13 | mpd3an3 1327 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
15 | 14 | eleq2d 2234 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀..^𝑁) ↔ 𝑥 ∈ (𝑀...(𝑁 − 1)))) |
16 | 15 | expcom 115 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑀 ∈ ℤ → (𝑥 ∈ (𝑀..^𝑁) ↔ 𝑥 ∈ (𝑀...(𝑁 − 1))))) |
17 | 2, 4, 16 | pm5.21ndd 695 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑥 ∈ (𝑀..^𝑁) ↔ 𝑥 ∈ (𝑀...(𝑁 − 1)))) |
18 | 17 | eqrdv 2162 | 1 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 𝒫 cpw 3553 (class class class)co 5836 1c1 7745 − cmin 8060 ℤcz 9182 ...cfz 9935 ..^cfzo 10067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-fzo 10068 |
This theorem is referenced by: elfzo 10074 fzodcel 10077 fzon 10091 fzoss1 10096 fzoss2 10097 fzval3 10129 fzo0to2pr 10143 fzo0to3tp 10144 fzo0to42pr 10145 fzoend 10147 fzofzp1b 10153 elfzom1b 10154 peano2fzor 10157 fzoshftral 10163 zmodfzo 10272 zmodidfzo 10278 fzofig 10357 hashfzo 10724 fzosump1 11344 telfsumo 11393 fsumparts 11397 geoserap 11434 geo2sum2 11442 dfphi2 12131 reumodprminv 12164 |
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