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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpexnle | Structured version Visualization version GIF version |
Description: There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.) |
Ref | Expression |
---|---|
lhp2a.l | ⊢ ≤ = (le‘𝐾) |
lhp2a.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhp2a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpexnle | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
2 | eqid 2795 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
3 | lhp2a.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhp1cvr 36666 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾)) |
5 | simpl 483 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ HL) | |
6 | eqid 2795 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
7 | 6, 3 | lhpbase 36665 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ (Base‘𝐾)) |
9 | hlop 36029 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | 6, 1 | op1cl 35852 | . . . . . 6 ⊢ (𝐾 ∈ OP → (1.‘𝐾) ∈ (Base‘𝐾)) |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝐾 ∈ HL → (1.‘𝐾) ∈ (Base‘𝐾)) |
12 | 11 | adantr 481 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1.‘𝐾) ∈ (Base‘𝐾)) |
13 | lhp2a.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
14 | eqid 2795 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
15 | lhp2a.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
16 | 6, 13, 14, 2, 15 | cvrval3 36080 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ (Base‘𝐾) ∧ (1.‘𝐾) ∈ (Base‘𝐾)) → (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑊(join‘𝐾)𝑝) = (1.‘𝐾)))) |
17 | 5, 8, 12, 16 | syl3anc 1364 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑊(join‘𝐾)𝑝) = (1.‘𝐾)))) |
18 | 4, 17 | mpbid 233 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑊(join‘𝐾)𝑝) = (1.‘𝐾))) |
19 | simpl 483 | . . 3 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ (𝑊(join‘𝐾)𝑝) = (1.‘𝐾)) → ¬ 𝑝 ≤ 𝑊) | |
20 | 19 | reximi 3207 | . 2 ⊢ (∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑊(join‘𝐾)𝑝) = (1.‘𝐾)) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊) |
21 | 18, 20 | syl 17 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 lecple 16401 joincjn 17383 1.cp1 17477 OPcops 35839 ⋖ ccvr 35929 Atomscatm 35930 HLchlt 36017 LHypclh 36651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-oposet 35843 df-ol 35845 df-oml 35846 df-covers 35933 df-ats 35934 df-atl 35965 df-cvlat 35989 df-hlat 36018 df-lhyp 36655 |
This theorem is referenced by: trlcnv 36832 trlator0 36838 trlid0 36843 trlnidatb 36844 cdlemf2 37229 cdlemg1cex 37255 trlco 37394 cdlemg44 37400 |
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