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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 13635 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
3 | simpl 481 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ (𝑀..^𝑁)) | |
4 | fzoval 13637 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
5 | 4 | adantl 480 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
6 | 3, 5 | eleqtrd 2833 | . . 3 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ (𝑀...(𝑁 − 1))) |
7 | elfzle2 13509 | . . 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 582 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 (class class class)co 7411 1c1 11113 ≤ cle 11253 − cmin 11448 ℤcz 12562 ...cfz 13488 ..^cfzo 13631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-neg 11451 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 |
This theorem is referenced by: iundjiun 45474 |
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