Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemedb | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 20-Nov-2012.) |
Ref | Expression |
---|---|
cdlemeda.l | ⊢ ≤ = (le‘𝐾) |
cdlemeda.j | ⊢ ∨ = (join‘𝐾) |
cdlemeda.m | ⊢ ∧ = (meet‘𝐾) |
cdlemeda.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemeda.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemeda.d | ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) |
cdlemedb.b | ⊢ 𝐵 = (Base‘𝐾) |
Ref | Expression |
---|---|
cdlemedb | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐷 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemeda.d | . 2 ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) | |
2 | hllat 33668 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 2 | ad2antrr 758 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat) |
4 | simpll 786 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL) | |
5 | simprl 790 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ 𝐴) | |
6 | simprr 792 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ 𝐴) | |
7 | cdlemedb.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
8 | cdlemeda.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
9 | cdlemeda.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 7, 8, 9 | hlatjcl 33671 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ 𝐵) |
11 | 4, 5, 6, 10 | syl3anc 1318 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑅 ∨ 𝑆) ∈ 𝐵) |
12 | cdlemeda.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
13 | 7, 12 | lhpbase 34302 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
14 | 13 | ad2antlr 759 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑊 ∈ 𝐵) |
15 | cdlemeda.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
16 | 7, 15 | latmcl 16875 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
17 | 3, 11, 14, 16 | syl3anc 1318 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
18 | 1, 17 | syl5eqel 2692 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐷 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Latclat 16868 Atomscatm 33568 HLchlt 33655 LHypclh 34288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-lat 16869 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-lhyp 34292 |
This theorem is referenced by: cdleme20k 34625 cdleme20l2 34627 cdleme20l 34628 cdleme20m 34629 |
Copyright terms: Public domain | W3C validator |