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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 10184 | . 2 ⊢ (𝐼 ∈ (0..^𝑁) ↔ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁)) | |
2 | peano2nn0 9218 | . . . 4 ⊢ (𝐼 ∈ ℕ0 → (𝐼 + 1) ∈ ℕ0) | |
3 | 2 | 3ad2ant1 1018 | . . 3 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 + 1) ∈ ℕ0) |
4 | peano2nn 8933 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | 3ad2ant2 1019 | . . 3 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝑁 + 1) ∈ ℕ) |
6 | simp3 999 | . . . 4 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → 𝐼 < 𝑁) | |
7 | nn0re 9187 | . . . . 5 ⊢ (𝐼 ∈ ℕ0 → 𝐼 ∈ ℝ) | |
8 | nnre 8928 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 7974 | . . . . 5 ⊢ (𝐼 < 𝑁 → 1 ∈ ℝ) | |
10 | ltadd1 8388 | . . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐼 < 𝑁 ↔ (𝐼 + 1) < (𝑁 + 1))) | |
11 | 7, 8, 9, 10 | syl3an 1280 | . . . 4 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 < 𝑁 ↔ (𝐼 + 1) < (𝑁 + 1))) |
12 | 6, 11 | mpbid 147 | . . 3 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 + 1) < (𝑁 + 1)) |
13 | elfzo0 10184 | . . 3 ⊢ ((𝐼 + 1) ∈ (0..^(𝑁 + 1)) ↔ ((𝐼 + 1) ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ ∧ (𝐼 + 1) < (𝑁 + 1))) | |
14 | 3, 5, 12, 13 | syl3anbrc 1181 | . 2 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1))) |
15 | 1, 14 | sylbi 121 | 1 ⊢ (𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 978 ∈ wcel 2148 class class class wbr 4005 (class class class)co 5877 ℝcr 7812 0cc0 7813 1c1 7814 + caddc 7816 < clt 7994 ℕcn 8921 ℕ0cn0 9178 ..^cfzo 10144 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 df-uz 9531 df-fz 10011 df-fzo 10145 |
This theorem is referenced by: (None) |
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