Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpjat1 | Structured version Visualization version GIF version |
Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
lhpjat.l | ⊢ ≤ = (le‘𝐾) |
lhpjat.j | ⊢ ∨ = (join‘𝐾) |
lhpjat.u | ⊢ 1 = (1.‘𝐾) |
lhpjat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpjat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpjat1 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 763 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL) | |
2 | eqid 2818 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lhpjat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | lhpbase 37014 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
5 | 4 | ad2antlr 723 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
6 | simprl 767 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) | |
7 | lhpjat.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
8 | eqid 2818 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | 7, 8, 3 | lhp1cvr 37015 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾) 1 ) |
10 | 9 | adantr 481 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊( ⋖ ‘𝐾) 1 ) |
11 | simprr 769 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ 𝑃 ≤ 𝑊) | |
12 | lhpjat.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | lhpjat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
14 | lhpjat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
15 | 2, 12, 13, 7, 8, 14 | 1cvrjat 36491 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ (Base‘𝐾) ∧ 𝑃 ∈ 𝐴) ∧ (𝑊( ⋖ ‘𝐾) 1 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
16 | 1, 5, 6, 10, 11, 15 | syl32anc 1370 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 1.cp1 17636 ⋖ ccvr 36278 Atomscatm 36279 HLchlt 36366 LHypclh 37000 |
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-p1 17638 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-lhyp 37004 |
This theorem is referenced by: lhpjat2 37037 lhpj1 37038 trljat1 37182 trljat2 37183 cdlemc1 37207 cdlemc6 37212 cdleme20c 37327 cdleme20j 37334 trlcolem 37742 |
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