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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolnltN | Structured version Visualization version GIF version |
Description: Two lattice volumes cannot satisfy the less than relation. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lvolnlt.s | ⊢ < = (lt‘𝐾) |
lvolnlt.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
lvolnltN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋 < 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvolnlt.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
2 | 1 | pltirr 17846 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 < 𝑋) |
3 | 2 | 3adant3 1134 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋 < 𝑋) |
4 | breq2 5062 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 < 𝑋 ↔ 𝑋 < 𝑌)) | |
5 | 4 | notbid 321 | . . 3 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 < 𝑋 ↔ ¬ 𝑋 < 𝑌)) |
6 | 3, 5 | syl5ibcom 248 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 = 𝑌 → ¬ 𝑋 < 𝑌)) |
7 | eqid 2737 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 17844 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 < 𝑌 → 𝑋(le‘𝐾)𝑌)) |
9 | lvolnlt.v | . . . . 5 ⊢ 𝑉 = (LVols‘𝐾) | |
10 | 7, 9 | lvolcmp 37373 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋(le‘𝐾)𝑌 ↔ 𝑋 = 𝑌)) |
11 | 8, 10 | sylibd 242 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 < 𝑌 → 𝑋 = 𝑌)) |
12 | 11 | necon3ad 2953 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≠ 𝑌 → ¬ 𝑋 < 𝑌)) |
13 | 6, 12 | pm2.61dne 3028 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋 < 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6385 lecple 16814 ltcplt 17820 HLchlt 37106 LVolsclvol 37249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-proset 17807 df-poset 17825 df-plt 17841 df-lub 17857 df-glb 17858 df-join 17859 df-meet 17860 df-p0 17936 df-lat 17943 df-clat 18010 df-oposet 36932 df-ol 36934 df-oml 36935 df-covers 37022 df-ats 37023 df-atl 37054 df-cvlat 37078 df-hlat 37107 df-llines 37254 df-lplanes 37255 df-lvols 37256 |
This theorem is referenced by: (None) |
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