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Mirrors > Home > ILE Home > Th. List > fzossrbm1 | GIF version |
Description: Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
Ref | Expression |
---|---|
fzossrbm1 | ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 9310 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | id 19 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
3 | zre 9276 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
4 | 3 | lem1d 8909 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ≤ 𝑁) |
5 | eluz2 9553 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) ↔ ((𝑁 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 − 1) ≤ 𝑁)) | |
6 | 1, 2, 4, 5 | syl3anbrc 1183 | . 2 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
7 | fzoss2 10191 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁)) | |
8 | 6, 7 | syl 14 | 1 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ⊆ wss 3144 class class class wbr 4018 ‘cfv 5231 (class class class)co 5891 0cc0 7830 1c1 7831 ≤ cle 8012 − cmin 8147 ℤcz 9272 ℤ≥cuz 9547 ..^cfzo 10161 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-ltadd 7946 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-inn 8939 df-n0 9196 df-z 9273 df-uz 9548 df-fz 10028 df-fzo 10162 |
This theorem is referenced by: elfzom1elp1fzo 10221 elfzom1elfzo 10222 |
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