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Mirrors > Home > ILE Home > Th. List > elfzom1p1elfzo | GIF version |
Description: Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) |
Ref | Expression |
---|---|
elfzom1p1elfzo | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo0 9191 | . . 3 ⊢ (𝑋 ∈ (0..^(𝑁 − 1)) ↔ (𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1))) | |
2 | peano2nn0 8328 | . . . . . . 7 ⊢ (𝑋 ∈ ℕ0 → (𝑋 + 1) ∈ ℕ0) | |
3 | 2 | 3ad2ant1 959 | . . . . . 6 ⊢ ((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) → (𝑋 + 1) ∈ ℕ0) |
4 | 3 | adantr 270 | . . . . 5 ⊢ (((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (𝑋 + 1) ∈ ℕ0) |
5 | simpr 108 | . . . . 5 ⊢ (((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
6 | nn0re 8297 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ ℕ0 → 𝑋 ∈ ℝ) | |
7 | 6 | adantr 270 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ ℝ) |
8 | 1red 7134 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → 1 ∈ ℝ) | |
9 | nnre 8046 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
10 | 9 | adantl 271 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ) |
11 | 7, 8, 10 | ltaddsubd 7645 | . . . . . . . . 9 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → ((𝑋 + 1) < 𝑁 ↔ 𝑋 < (𝑁 − 1))) |
12 | 11 | biimprd 156 | . . . . . . . 8 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑋 < (𝑁 − 1) → (𝑋 + 1) < 𝑁)) |
13 | 12 | impancom 256 | . . . . . . 7 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑋 < (𝑁 − 1)) → (𝑁 ∈ ℕ → (𝑋 + 1) < 𝑁)) |
14 | 13 | 3adant2 957 | . . . . . 6 ⊢ ((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) → (𝑁 ∈ ℕ → (𝑋 + 1) < 𝑁)) |
15 | 14 | imp 122 | . . . . 5 ⊢ (((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (𝑋 + 1) < 𝑁) |
16 | elfzo0 9191 | . . . . 5 ⊢ ((𝑋 + 1) ∈ (0..^𝑁) ↔ ((𝑋 + 1) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑋 + 1) < 𝑁)) | |
17 | 4, 5, 15, 16 | syl3anbrc 1122 | . . . 4 ⊢ (((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (𝑋 + 1) ∈ (0..^𝑁)) |
18 | 17 | ex 113 | . . 3 ⊢ ((𝑋 ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ ∧ 𝑋 < (𝑁 − 1)) → (𝑁 ∈ ℕ → (𝑋 + 1) ∈ (0..^𝑁))) |
19 | 1, 18 | sylbi 119 | . 2 ⊢ (𝑋 ∈ (0..^(𝑁 − 1)) → (𝑁 ∈ ℕ → (𝑋 + 1) ∈ (0..^𝑁))) |
20 | 19 | impcom 123 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 1c1 6982 + caddc 6984 < clt 7153 − cmin 7279 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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