Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpjat2 | 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, 4-Jun-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 |
---|---|
lhpjat2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 36501 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | ad2antrr 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat) |
3 | eqid 2823 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | lhpjat.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 36427 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 5 | ad2antrl 726 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ (Base‘𝐾)) |
7 | lhpjat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | 3, 7 | lhpbase 37136 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
9 | 8 | ad2antlr 725 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
10 | lhpjat.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
11 | 3, 10 | latjcom 17671 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑊) = (𝑊 ∨ 𝑃)) |
12 | 2, 6, 9, 11 | syl3anc 1367 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (𝑊 ∨ 𝑃)) |
13 | lhpjat.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
14 | lhpjat.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
15 | 13, 10, 14, 4, 7 | lhpjat1 37158 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
16 | 12, 15 | eqtrd 2858 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 1.cp1 17650 Latclat 17657 Atomscatm 36401 HLchlt 36488 LHypclh 37122 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 |
This theorem is referenced by: lhpmcvr3 37163 cdleme0cp 37352 cdleme0cq 37353 cdleme1 37365 cdleme4 37376 cdleme5 37378 cdleme8 37388 cdleme9 37391 cdleme10 37392 cdleme22e 37482 cdleme22eALTN 37483 cdleme35b 37588 cdleme35e 37591 cdleme42a 37609 trlcoabs2N 37860 cdlemi1 37956 cdlemk4 37972 dia2dimlem1 38202 cdlemn10 38344 dihglbcpreN 38438 |
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