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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpat | Structured version Visualization version GIF version |
Description: Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.) |
Ref | Expression |
---|---|
lhpat.l | ⊢ ≤ = (le‘𝐾) |
lhpat.j | ⊢ ∨ = (join‘𝐾) |
lhpat.m | ⊢ ∧ = (meet‘𝐾) |
lhpat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1255 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝐾 ∈ HL) | |
2 | simp2l 1257 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ 𝐴) | |
3 | simp3l 1259 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ 𝐴) | |
4 | simp1r 1256 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑊 ∈ 𝐻) | |
5 | eqid 2797 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | lhpat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | 5, 6 | lhpbase 36010 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
8 | 4, 7 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑊 ∈ (Base‘𝐾)) |
9 | simp3r 1260 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑃 ≠ 𝑄) | |
10 | eqid 2797 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
11 | eqid 2797 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
12 | 10, 11, 6 | lhp1cvr 36011 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾)) |
13 | 12 | 3ad2ant1 1164 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾)) |
14 | simp2r 1258 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ¬ 𝑃 ≤ 𝑊) | |
15 | lhpat.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
16 | lhpat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
17 | lhpat.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
18 | lhpat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
19 | 5, 15, 16, 17, 10, 11, 18 | 1cvrat 35488 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑊( ⋖ ‘𝐾)(1.‘𝐾) ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) |
20 | 1, 2, 3, 8, 9, 13, 14, 19 | syl133anc 1513 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 class class class wbr 4841 ‘cfv 6099 (class class class)co 6876 Basecbs 16180 lecple 16270 joincjn 17255 meetcmee 17256 1.cp1 17349 ⋖ ccvr 35274 Atomscatm 35275 HLchlt 35362 LHypclh 35996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-proset 17239 df-poset 17257 df-plt 17269 df-lub 17285 df-glb 17286 df-join 17287 df-meet 17288 df-p0 17350 df-p1 17351 df-lat 17357 df-clat 17419 df-oposet 35188 df-ol 35190 df-oml 35191 df-covers 35278 df-ats 35279 df-atl 35310 df-cvlat 35334 df-hlat 35363 df-lhyp 36000 |
This theorem is referenced by: lhpat2 36057 4atexlemex6 36086 trlat 36181 cdlemc5 36207 cdleme3e 36244 cdleme7b 36256 cdleme11k 36280 cdleme16e 36294 cdleme16f 36295 cdlemeda 36310 cdleme22cN 36354 cdleme22d 36355 cdleme23b 36362 cdlemf2 36574 cdlemg12g 36661 cdlemg17dALTN 36676 cdlemg19a 36695 |
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