Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 1185 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝑌 ∈ 𝑃) | |
2 | simpl1 1183 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝐾 ∈ HL) | |
3 | hllat 36379 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
4 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 2lplnm.p | . . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾) | |
6 | 4, 5 | lplnbase 36550 | . . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
7 | 4, 5 | lplnbase 36550 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → 𝑌 ∈ (Base‘𝐾)) |
8 | 2lplnm.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
9 | 4, 8 | latmcl 17650 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
10 | 3, 6, 7, 9 | syl3an 1152 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
11 | 10 | adantr 481 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
12 | 7 | 3ad2ant3 1127 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑌 ∈ (Base‘𝐾)) |
13 | 12 | adantr 481 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝑌 ∈ (Base‘𝐾)) |
14 | simp1 1128 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ HL) | |
15 | 6 | 3ad2ant2 1126 | . . . . 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 36436 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌))) |
19 | 14, 15, 12, 18 | syl3anc 1363 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌))) |
20 | 19 | biimpar 478 | . . 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 36574 | . . 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 1365 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → ((𝑋 ∧ 𝑌) ∈ 𝑁 ↔ 𝑌 ∈ 𝑃)) |
24 | 1, 23 | mpbird 258 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 joincjn 17542 meetcmee 17543 Latclat 17643 ⋖ ccvr 36278 HLchlt 36366 LLinesclln 36507 LPlanesclpl 36508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |