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Mirrors > Home > ILE Home > Th. List > fzonn0p1p1 | GIF version |
Description: If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
Ref | Expression |
---|---|
fzonn0p1p1 | ⊢ (𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo0 9191 | . 2 ⊢ (𝐼 ∈ (0..^𝑁) ↔ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁)) | |
2 | peano2nn0 8328 | . . . 4 ⊢ (𝐼 ∈ ℕ0 → (𝐼 + 1) ∈ ℕ0) | |
3 | 2 | 3ad2ant1 959 | . . 3 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 + 1) ∈ ℕ0) |
4 | peano2nn 8051 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | 3ad2ant2 960 | . . 3 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝑁 + 1) ∈ ℕ) |
6 | simp3 940 | . . . 4 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → 𝐼 < 𝑁) | |
7 | nn0re 8297 | . . . . 5 ⊢ (𝐼 ∈ ℕ0 → 𝐼 ∈ ℝ) | |
8 | nnre 8046 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 7134 | . . . . 5 ⊢ (𝐼 < 𝑁 → 1 ∈ ℝ) | |
10 | ltadd1 7533 | . . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐼 < 𝑁 ↔ (𝐼 + 1) < (𝑁 + 1))) | |
11 | 7, 8, 9, 10 | syl3an 1211 | . . . 4 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 < 𝑁 ↔ (𝐼 + 1) < (𝑁 + 1))) |
12 | 6, 11 | mpbid 145 | . . 3 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 + 1) < (𝑁 + 1)) |
13 | elfzo0 9191 | . . 3 ⊢ ((𝐼 + 1) ∈ (0..^(𝑁 + 1)) ↔ ((𝐼 + 1) ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ ∧ (𝐼 + 1) < (𝑁 + 1))) | |
14 | 3, 5, 12, 13 | syl3anbrc 1122 | . 2 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1))) |
15 | 1, 14 | sylbi 119 | 1 ⊢ (𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 1c1 6982 + caddc 6984 < clt 7153 ℕcn 8039 ℕ0cn0 8288 ..^cfzo 9152 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 df-fzo 9153 |
This theorem is referenced by: (None) |
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