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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 9953 | . . . 4 ⊢ (𝑥 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑥 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)) |
3 | elfzel1 9836 | . . . 4 ⊢ (𝑥 ∈ (𝑀...(𝑁 − 1)) → 𝑀 ∈ ℤ) | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑥 ∈ (𝑀...(𝑁 − 1)) → 𝑀 ∈ ℤ)) |
5 | peano2zm 9116 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
6 | fzf 9825 | . . . . . . . 8 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
7 | 6 | fovcl 5884 | . . . . . . 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 5789 | . . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝑧 − 1) = (𝑁 − 1)) | |
11 | 9, 10 | oveqan12d 5801 | . . . . . . 7 ⊢ ((𝑦 = 𝑀 ∧ 𝑧 = 𝑁) → (𝑦...(𝑧 − 1)) = (𝑀...(𝑁 − 1))) |
12 | df-fzo 9951 | . . . . . . 7 ⊢ ..^ = (𝑦 ∈ ℤ, 𝑧 ∈ ℤ ↦ (𝑦...(𝑧 − 1))) | |
13 | 11, 12 | ovmpoga 5908 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀...(𝑁 − 1)) ∈ 𝒫 ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
14 | 8, 13 | mpd3an3 1317 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
15 | 14 | eleq2d 2210 | . . . 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 2138 | 1 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 𝒫 cpw 3515 (class class class)co 5782 1c1 7645 − cmin 7957 ℤcz 9078 ...cfz 9821 ..^cfzo 9950 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-fzo 9951 |
This theorem is referenced by: elfzo 9957 fzodcel 9960 fzon 9974 fzoss1 9979 fzoss2 9980 fzval3 10012 fzo0to2pr 10026 fzo0to3tp 10027 fzo0to42pr 10028 fzoend 10030 fzofzp1b 10036 elfzom1b 10037 peano2fzor 10040 fzoshftral 10046 zmodfzo 10151 zmodidfzo 10157 fzofig 10236 hashfzo 10600 fzosump1 11218 telfsumo 11267 fsumparts 11271 geoserap 11308 geo2sum2 11316 dfphi2 11932 |
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