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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2lplnmN | Structured version Visualization version GIF version |
Description: If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2lplnm.j | ⊢ ∨ = (join‘𝐾) |
2lplnm.m | ⊢ ∧ = (meet‘𝐾) |
2lplnm.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
2lplnm.n | ⊢ 𝑁 = (LLines‘𝐾) |
2lplnm.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
2lplnmN | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1174 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝑌 ∈ 𝑃) | |
2 | simpl1 1172 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝐾 ∈ HL) | |
3 | hllat 35977 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
4 | eqid 2773 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 2lplnm.p | . . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾) | |
6 | 4, 5 | lplnbase 36148 | . . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
7 | 4, 5 | lplnbase 36148 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → 𝑌 ∈ (Base‘𝐾)) |
8 | 2lplnm.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
9 | 4, 8 | latmcl 17533 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
10 | 3, 6, 7, 9 | syl3an 1141 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
11 | 10 | adantr 473 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
12 | 7 | 3ad2ant3 1116 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑌 ∈ (Base‘𝐾)) |
13 | 12 | adantr 473 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝑌 ∈ (Base‘𝐾)) |
14 | simp1 1117 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ HL) | |
15 | 6 | 3ad2ant2 1115 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ∈ (Base‘𝐾)) |
16 | 2lplnm.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
17 | 2lplnm.c | . . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
18 | 4, 16, 8, 17 | cvrexch 36034 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌))) |
19 | 14, 15, 12, 18 | syl3anc 1352 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌))) |
20 | 19 | biimpar 470 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌)𝐶𝑌) |
21 | 2lplnm.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
22 | 4, 17, 21, 5 | llncvrlpln 36172 | . . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑋 ∧ 𝑌)𝐶𝑌) → ((𝑋 ∧ 𝑌) ∈ 𝑁 ↔ 𝑌 ∈ 𝑃)) |
23 | 2, 11, 13, 20, 22 | syl31anc 1354 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → ((𝑋 ∧ 𝑌) ∈ 𝑁 ↔ 𝑌 ∈ 𝑃)) |
24 | 1, 23 | mpbird 249 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 Basecbs 16338 joincjn 17425 meetcmee 17426 Latclat 17526 ⋖ ccvr 35876 HLchlt 35964 LLinesclln 36105 LPlanesclpl 36106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-proset 17409 df-poset 17427 df-plt 17439 df-lub 17455 df-glb 17456 df-join 17457 df-meet 17458 df-p0 17520 df-lat 17527 df-clat 17589 df-oposet 35790 df-ol 35792 df-oml 35793 df-covers 35880 df-ats 35881 df-atl 35912 df-cvlat 35936 df-hlat 35965 df-llines 36112 df-lplanes 36113 |
This theorem is referenced by: (None) |
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