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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfzolem1 | Structured version Visualization version GIF version |
Description: A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
elfzolem1 | ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀..^𝑁)) | |
2 | elfzoel2 13596 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
3 | simpl 483 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ (𝑀..^𝑁)) | |
4 | fzoval 13598 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
6 | 3, 5 | eleqtrd 2834 | . . 3 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ (𝑀...(𝑁 − 1))) |
7 | elfzle2 13470 | . . 3 ⊢ (𝐾 ∈ (𝑀...(𝑁 − 1)) → 𝐾 ≤ (𝑁 − 1)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ≤ (𝑁 − 1)) |
9 | 1, 2, 8 | syl2anc 584 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5125 (class class class)co 7377 1c1 11076 ≤ cle 11214 − cmin 11409 ℤcz 12523 ...cfz 13449 ..^cfzo 13592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-1st 7941 df-2nd 7942 df-neg 11412 df-z 12524 df-uz 12788 df-fz 13450 df-fzo 13593 |
This theorem is referenced by: iundjiun 44854 |
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