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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 34650 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 2 | adantr 481 | . . 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 34653 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ 𝐵) |
8 | 7 | 3adant3r3 1276 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → (𝑃 ∨ 𝑆) ∈ 𝐵) |
9 | simpr3 1069 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝑊 ∈ 𝐻) | |
10 | cdleme9b.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | 4, 10 | lhpbase 35284 | . . . 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 17052 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
15 | 3, 8, 12, 14 | syl3anc 1326 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
16 | 1, 15 | syl5eqel 2705 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐶 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 joincjn 16944 meetcmee 16945 Latclat 17045 Atomscatm 34550 HLchlt 34637 LHypclh 35270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-lhyp 35274 |
This theorem is referenced by: cdleme15b 35562 cdleme17b 35574 |
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