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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzssfzo | Structured version Visualization version GIF version |
Description: Condition for an integer interval to be a subset of a half-open integer interval. (Contributed by Thierry Arnoux, 8-Oct-2018.) |
Ref | Expression |
---|---|
fzssfzo | ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀...𝐾) ⊆ (𝑀..^𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoel2 13038 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
2 | fzoval 13040 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
4 | 3 | eleq2d 2898 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 ∈ (𝑀..^𝑁) ↔ 𝐾 ∈ (𝑀...(𝑁 − 1)))) |
5 | 4 | ibi 269 | . . 3 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...(𝑁 − 1))) |
6 | elfzuz3 12906 | . . 3 ⊢ (𝐾 ∈ (𝑀...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ≥‘𝐾)) | |
7 | fzss2 12948 | . . 3 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...(𝑁 − 1))) | |
8 | 5, 6, 7 | 3syl 18 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀...𝐾) ⊆ (𝑀...(𝑁 − 1))) |
9 | 8, 3 | sseqtrrd 4008 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀...𝐾) ⊆ (𝑀..^𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 1c1 10538 − cmin 10870 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 ..^cfzo 13034 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
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 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: signstcl 31835 signstf 31836 signstfvp 31841 |
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