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Mirrors > Home > MPE Home > Th. List > fzofzp1b | Structured version Visualization version 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 13119 | . 2 ⊢ (𝐶 ∈ (𝐴..^𝐵) → (𝐶 + 1) ∈ (𝐴...𝐵)) | |
2 | simpl 485 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐶 ∈ (ℤ≥‘𝐴)) | |
3 | eluzelz 12235 | . . . . . 6 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → 𝐶 ∈ ℤ) | |
4 | elfzuz3 12890 | . . . . . 6 ⊢ ((𝐶 + 1) ∈ (𝐴...𝐵) → 𝐵 ∈ (ℤ≥‘(𝐶 + 1))) | |
5 | eluzp1m1 12250 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐶 + 1))) → (𝐵 − 1) ∈ (ℤ≥‘𝐶)) | |
6 | 3, 4, 5 | syl2an 597 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → (𝐵 − 1) ∈ (ℤ≥‘𝐶)) |
7 | elfzuzb 12887 | . . . . 5 ⊢ (𝐶 ∈ (𝐴...(𝐵 − 1)) ↔ (𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐵 − 1) ∈ (ℤ≥‘𝐶))) | |
8 | 2, 6, 7 | sylanbrc 585 | . . . 4 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐶 ∈ (𝐴...(𝐵 − 1))) |
9 | elfzel2 12891 | . . . . . 6 ⊢ ((𝐶 + 1) ∈ (𝐴...𝐵) → 𝐵 ∈ ℤ) | |
10 | 9 | adantl 484 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐵 ∈ ℤ) |
11 | fzoval 13024 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) |
13 | 8, 12 | eleqtrrd 2914 | . . 3 ⊢ ((𝐶 ∈ (ℤ≥‘𝐴) ∧ (𝐶 + 1) ∈ (𝐴...𝐵)) → 𝐶 ∈ (𝐴..^𝐵)) |
14 | 13 | ex 415 | . 2 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → ((𝐶 + 1) ∈ (𝐴...𝐵) → 𝐶 ∈ (𝐴..^𝐵))) |
15 | 1, 14 | impbid2 228 | 1 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6336 (class class class)co 7137 1c1 10519 + caddc 10521 − cmin 10851 ℤcz 11963 ℤ≥cuz 12225 ...cfz 12877 ..^cfzo 13018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-n0 11880 df-z 11964 df-uz 12226 df-fz 12878 df-fzo 13019 |
This theorem is referenced by: iccpartres 43663 |
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