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Mirrors > Home > ILE Home > Th. List > ubmelfzo | GIF version |
Description: If an integer in a 1 based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.) |
Ref | Expression |
---|---|
ubmelfzo | ⊢ (𝐾 ∈ (1...𝑁) → (𝑁 − 𝐾) ∈ (0..^𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 941 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → 𝐾 ≤ 𝑁) | |
2 | nnnn0 8432 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ0) | |
3 | nnnn0 8432 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | anim12i 331 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) |
5 | 4 | 3adant3 959 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) |
6 | nn0sub 8568 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐾 ≤ 𝑁 ↔ (𝑁 − 𝐾) ∈ ℕ0)) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → (𝐾 ≤ 𝑁 ↔ (𝑁 − 𝐾) ∈ ℕ0)) |
8 | 1, 7 | mpbid 145 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → (𝑁 − 𝐾) ∈ ℕ0) |
9 | simp2 940 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → 𝑁 ∈ ℕ) | |
10 | nngt0 8201 | . . . . 5 ⊢ (𝐾 ∈ ℕ → 0 < 𝐾) | |
11 | 10 | 3ad2ant1 960 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → 0 < 𝐾) |
12 | nnre 8183 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℝ) | |
13 | nnre 8183 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
14 | 12, 13 | anim12i 331 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
15 | 14 | 3adant3 959 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → (𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
16 | ltsubpos 7695 | . . . . 5 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝐾 ↔ (𝑁 − 𝐾) < 𝑁)) | |
17 | 15, 16 | syl 14 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → (0 < 𝐾 ↔ (𝑁 − 𝐾) < 𝑁)) |
18 | 11, 17 | mpbid 145 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → (𝑁 − 𝐾) < 𝑁) |
19 | 8, 9, 18 | 3jca 1119 | . 2 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁) → ((𝑁 − 𝐾) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑁 − 𝐾) < 𝑁)) |
20 | elfz1b 9253 | . 2 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁)) | |
21 | elfzo0 9338 | . 2 ⊢ ((𝑁 − 𝐾) ∈ (0..^𝑁) ↔ ((𝑁 − 𝐾) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑁 − 𝐾) < 𝑁)) | |
22 | 19, 20, 21 | 3imtr4i 199 | 1 ⊢ (𝐾 ∈ (1...𝑁) → (𝑁 − 𝐾) ∈ (0..^𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 920 ∈ wcel 1434 class class class wbr 3805 (class class class)co 5564 ℝcr 7112 0cc0 7113 1c1 7114 < clt 7285 ≤ cle 7286 − cmin 7416 ℕcn 8176 ℕ0cn0 8425 ...cfz 9175 ..^cfzo 9299 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7199 ax-resscn 7200 ax-1cn 7201 ax-1re 7202 ax-icn 7203 ax-addcl 7204 ax-addrcl 7205 ax-mulcl 7206 ax-addcom 7208 ax-addass 7210 ax-distr 7212 ax-i2m1 7213 ax-0lt1 7214 ax-0id 7216 ax-rnegex 7217 ax-cnre 7219 ax-pre-ltirr 7220 ax-pre-ltwlin 7221 ax-pre-lttrn 7222 ax-pre-ltadd 7224 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-id 4076 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-fv 4960 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-1st 5819 df-2nd 5820 df-pnf 7287 df-mnf 7288 df-xr 7289 df-ltxr 7290 df-le 7291 df-sub 7418 df-neg 7419 df-inn 8177 df-n0 8426 df-z 8503 df-uz 8771 df-fz 9176 df-fzo 9300 |
This theorem is referenced by: (None) |
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