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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2llnm4 | Structured version Visualization version GIF version |
Description: Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.) |
Ref | Expression |
---|---|
2llnm4.l | ⊢ ≤ = (le‘𝐾) |
2llnm4.m | ⊢ ∧ = (meet‘𝐾) |
2llnm4.z | ⊢ 0 = (0.‘𝐾) |
2llnm4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
2llnm4.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
2llnm4 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatl 36511 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
2 | 1 | 3ad2ant1 1129 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝐾 ∈ AtLat) |
3 | hllat 36514 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
4 | 3 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝐾 ∈ Lat) |
5 | simp22 1203 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑋 ∈ 𝑁) | |
6 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
7 | 2llnm4.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
8 | 6, 7 | llnbase 36660 | . . . 4 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾)) |
9 | 5, 8 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾)) |
10 | simp23 1204 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑌 ∈ 𝑁) | |
11 | 6, 7 | llnbase 36660 | . . . 4 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾)) |
12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾)) |
13 | 2llnm4.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
14 | 6, 13 | latmcl 17662 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
15 | 4, 9, 12, 14 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
16 | simp21 1202 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑃 ∈ 𝐴) | |
17 | simp3 1134 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) | |
18 | 2llnm4.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
19 | 6, 18 | atbase 36440 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
20 | 16, 19 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑃 ∈ (Base‘𝐾)) |
21 | 2llnm4.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
22 | 6, 21, 13 | latlem12 17688 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌) ↔ 𝑃 ≤ (𝑋 ∧ 𝑌))) |
23 | 4, 20, 9, 12, 22 | syl13anc 1368 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → ((𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌) ↔ 𝑃 ≤ (𝑋 ∧ 𝑌))) |
24 | 17, 23 | mpbid 234 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑃 ≤ (𝑋 ∧ 𝑌)) |
25 | 2llnm4.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
26 | 6, 21, 25, 18 | atlen0 36461 | . 2 ⊢ (((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾) ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ (𝑋 ∧ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 ) |
27 | 2, 15, 16, 24, 26 | syl31anc 1369 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 lecple 16572 meetcmee 17555 0.cp0 17647 Latclat 17655 Atomscatm 36414 AtLatcal 36415 HLchlt 36501 LLinesclln 36642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-lat 17656 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 |
This theorem is referenced by: 2llnmeqat 36722 dalem2 36812 |
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