![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 13695 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
2 | fzoval 13697 | . . . . . 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 2825 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 ∈ (𝑀..^𝑁) ↔ 𝐾 ∈ (𝑀...(𝑁 − 1)))) |
5 | 4 | ibi 267 | . . 3 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...(𝑁 − 1))) |
6 | elfzuz3 13558 | . . 3 ⊢ (𝐾 ∈ (𝑀...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ≥‘𝐾)) | |
7 | fzss2 13601 | . . 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 4037 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀...𝐾) ⊆ (𝑀..^𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 1c1 11154 − cmin 11490 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 ..^cfzo 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 |
This theorem is referenced by: signstcl 34559 signstf 34560 signstfvp 34565 |
Copyright terms: Public domain | W3C validator |