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Mirrors > Home > ILE Home > Th. List > fzofzp1b | GIF version |
Description: If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
fzofzp1b | ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzofzp1 10245 | . 2 ⊢ (𝐶 ∈ (𝐴..^𝐵) → (𝐶 + 1) ∈ (𝐴...𝐵)) | |
2 | simpl 109 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐶 ∈ (ℤ≥‘𝐴)) | |
3 | eluzelz 9555 | . . . . . 6 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → 𝐶 ∈ ℤ) | |
4 | elfzuz3 10040 | . . . . . 6 ⊢ ((𝐶 + 1) ∈ (𝐴...𝐵) → 𝐵 ∈ (ℤ≥‘(𝐶 + 1))) | |
5 | eluzp1m1 9569 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐶 + 1))) → (𝐵 − 1) ∈ (ℤ≥‘𝐶)) | |
6 | 3, 4, 5 | syl2an 289 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → (𝐵 − 1) ∈ (ℤ≥‘𝐶)) |
7 | elfzuzb 10037 | . . . . 5 ⊢ (𝐶 ∈ (𝐴...(𝐵 − 1)) ↔ (𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐵 − 1) ∈ (ℤ≥‘𝐶))) | |
8 | 2, 6, 7 | sylanbrc 417 | . . . 4 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐶 ∈ (𝐴...(𝐵 − 1))) |
9 | elfzel2 10041 | . . . . . 6 ⊢ ((𝐶 + 1) ∈ (𝐴...𝐵) → 𝐵 ∈ ℤ) | |
10 | 9 | adantl 277 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐵 ∈ ℤ) |
11 | fzoval 10166 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) | |
12 | 10, 11 | syl 14 | . . . 4 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) |
13 | 8, 12 | eleqtrrd 2269 | . . 3 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐶 ∈ (𝐴..^𝐵)) |
14 | 13 | ex 115 | . 2 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → ((𝐶 + 1) ∈ (𝐴...𝐵) → 𝐶 ∈ (𝐴..^𝐵))) |
15 | 1, 14 | impbid2 143 | 1 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ‘cfv 5231 (class class class)co 5891 1c1 7830 + caddc 7832 − cmin 8146 ℤcz 9271 ℤ≥cuz 9546 ...cfz 10026 ..^cfzo 10160 |
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 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-ltadd 7945 |
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 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-inn 8938 df-n0 9195 df-z 9272 df-uz 9547 df-fz 10027 df-fzo 10161 |
This theorem is referenced by: (None) |
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