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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 12723 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
3 | simpl 475 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ (𝑀..^𝑁)) | |
4 | fzoval 12725 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
5 | 4 | adantl 474 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
6 | 3, 5 | eleqtrd 2881 | . . 3 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ (𝑀...(𝑁 − 1))) |
7 | elfzle2 12598 | . . 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 580 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 class class class wbr 4844 (class class class)co 6879 1c1 10226 ≤ cle 10365 − cmin 10557 ℤcz 11665 ...cfz 12579 ..^cfzo 12719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-1st 7402 df-2nd 7403 df-neg 10560 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 |
This theorem is referenced by: iundjiun 41415 |
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