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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpmcvr | Structured version Visualization version GIF version |
Description: The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.) |
Ref | Expression |
---|---|
lhpmcvr.b | ⊢ 𝐵 = (Base‘𝐾) |
lhpmcvr.l | ⊢ ≤ = (le‘𝐾) |
lhpmcvr.m | ⊢ ∧ = (meet‘𝐾) |
lhpmcvr.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
lhpmcvr.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpmcvr | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑋 ∧ 𝑊)𝐶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 39345 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | ad2antrr 726 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝐾 ∈ Lat) |
3 | simprl 771 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑋 ∈ 𝐵) | |
4 | lhpmcvr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | lhpmcvr.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | 4, 5 | lhpbase 39981 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
7 | 6 | ad2antlr 727 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑊 ∈ 𝐵) |
8 | lhpmcvr.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
9 | 4, 8 | latmcom 18521 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) = (𝑊 ∧ 𝑋)) |
10 | 2, 3, 7, 9 | syl3anc 1370 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑋 ∧ 𝑊) = (𝑊 ∧ 𝑋)) |
11 | eqid 2735 | . . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
12 | lhpmcvr.c | . . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
13 | 11, 12, 5 | lhp1cvr 39982 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊𝐶(1.‘𝐾)) |
14 | 13 | adantr 480 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑊𝐶(1.‘𝐾)) |
15 | lhpmcvr.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
16 | eqid 2735 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
17 | 4, 15, 16, 11, 5 | lhpj1 40005 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊(join‘𝐾)𝑋) = (1.‘𝐾)) |
18 | 14, 17 | breqtrrd 5176 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑊𝐶(𝑊(join‘𝐾)𝑋)) |
19 | simpll 767 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝐾 ∈ HL) | |
20 | 4, 16, 8, 12 | cvrexch 39403 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑊 ∧ 𝑋)𝐶𝑋 ↔ 𝑊𝐶(𝑊(join‘𝐾)𝑋))) |
21 | 19, 7, 3, 20 | syl3anc 1370 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ((𝑊 ∧ 𝑋)𝐶𝑋 ↔ 𝑊𝐶(𝑊(join‘𝐾)𝑋))) |
22 | 18, 21 | mpbird 257 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∧ 𝑋)𝐶𝑋) |
23 | 10, 22 | eqbrtrd 5170 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑋 ∧ 𝑊)𝐶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 meetcmee 18370 1.cp1 18482 Latclat 18489 ⋖ ccvr 39244 HLchlt 39332 LHypclh 39967 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 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-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-lhyp 39971 |
This theorem is referenced by: lhpmcvr2 40007 lhpm0atN 40012 |
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