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Mirrors > Home > ILE Home > Th. List > elfzo0 | GIF version |
Description: Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
elfzo0 | ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzouz 10217 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ (ℤ≥‘0)) | |
2 | elnn0uz 9630 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℤ≥‘0)) | |
3 | 1, 2 | sylibr 134 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ ℕ0) |
4 | elfzolt3b 10226 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 0 ∈ (0..^𝐵)) | |
5 | lbfzo0 10248 | . . . 4 ⊢ (0 ∈ (0..^𝐵) ↔ 𝐵 ∈ ℕ) | |
6 | 4, 5 | sylib 122 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐵 ∈ ℕ) |
7 | elfzolt2 10223 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 < 𝐵) | |
8 | 3, 6, 7 | 3jca 1179 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
9 | simp1 999 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
10 | 9, 2 | sylib 122 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (ℤ≥‘0)) |
11 | nnz 9336 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
12 | 11 | 3ad2ant2 1021 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℤ) |
13 | simp3 1001 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
14 | elfzo2 10216 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ (ℤ≥‘0) ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) | |
15 | 10, 12, 13, 14 | syl3anbrc 1183 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (0..^𝐵)) |
16 | 8, 15 | impbii 126 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∧ w3a 980 ∈ wcel 2164 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 0cc0 7872 < clt 8054 ℕcn 8982 ℕ0cn0 9240 ℤcz 9317 ℤ≥cuz 9592 ..^cfzo 10208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-fz 10075 df-fzo 10209 |
This theorem is referenced by: fzo1fzo0n0 10250 elfzo0z 10251 elfzo0le 10252 fzonmapblen 10254 fzofzim 10255 ubmelfzo 10267 elfzodifsumelfzo 10268 elfzonlteqm1 10277 fzonn0p1 10278 fzonn0p1p1 10280 elfzom1p1elfzo 10281 ubmelm1fzo 10293 subfzo0 10309 zmodidfzoimp 10425 modfzo0difsn 10466 modsumfzodifsn 10467 addmodlteq 10469 addmodlteqALT 12001 hashgcdlem 12376 |
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