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Mathbox for Norm Megill |
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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 34968 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 2 | ad2antrr 762 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat) |
4 | simpll 805 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL) | |
5 | simprl 809 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ 𝐴) | |
6 | simprr 811 | . . . 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 34971 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ 𝐵) |
11 | 4, 5, 6, 10 | syl3anc 1366 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑅 ∨ 𝑆) ∈ 𝐵) |
12 | cdlemeda.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
13 | 7, 12 | lhpbase 35602 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
14 | 13 | ad2antlr 763 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑊 ∈ 𝐵) |
15 | cdlemeda.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
16 | 7, 15 | latmcl 17099 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
17 | 3, 11, 14, 16 | syl3anc 1366 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
18 | 1, 17 | syl5eqel 2734 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐷 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 lecple 15995 joincjn 16991 meetcmee 16992 Latclat 17092 Atomscatm 34868 HLchlt 34955 LHypclh 35588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-lat 17093 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-lhyp 35592 |
This theorem is referenced by: cdleme20k 35924 cdleme20l2 35926 cdleme20l 35927 cdleme20m 35928 |
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