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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme9b | Structured version Visualization version GIF version |
Description: Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.) |
Ref | Expression |
---|---|
cdleme9b.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme9b.j | ⊢ ∨ = (join‘𝐾) |
cdleme9b.m | ⊢ ∧ = (meet‘𝐾) |
cdleme9b.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme9b.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme9b.c | ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme9b | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐶 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme9b.c | . 2 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊) | |
2 | hllat 35984 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 2 | adantr 473 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐾 ∈ Lat) |
4 | cdleme9b.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | cdleme9b.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | cdleme9b.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 4, 5, 6 | hlatjcl 35988 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ 𝐵) |
8 | 7 | 3adant3r3 1165 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → (𝑃 ∨ 𝑆) ∈ 𝐵) |
9 | simpr3 1177 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝑊 ∈ 𝐻) | |
10 | cdleme9b.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | 4, 10 | lhpbase 36619 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝑊 ∈ 𝐵) |
13 | cdleme9b.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
14 | 4, 13 | latmcl 17532 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
15 | 3, 8, 12, 14 | syl3anc 1352 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
16 | 1, 15 | syl5eqel 2872 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐶 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ‘cfv 6193 (class class class)co 6982 Basecbs 16345 joincjn 17424 meetcmee 17425 Latclat 17525 Atomscatm 35884 HLchlt 35971 LHypclh 36605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-lat 17526 df-ats 35888 df-atl 35919 df-cvlat 35943 df-hlat 35972 df-lhyp 36609 |
This theorem is referenced by: cdleme15b 36896 cdleme17b 36908 |
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