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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 39870 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
| 3 | 2 | ad2antrr 733 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat) |
| 4 | simpll 773 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL) | |
| 5 | simprl 777 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ 𝐴) | |
| 6 | simprr 779 | . . . 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 39874 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ 𝐵) |
| 11 | 4, 5, 6, 10 | syl3anc 1380 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑅 ∨ 𝑆) ∈ 𝐵) |
| 12 | cdlemeda.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 13 | 7, 12 | lhpbase 40505 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
| 14 | 13 | ad2antlr 734 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑊 ∈ 𝐵) |
| 15 | cdlemeda.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
| 16 | 7, 15 | latmcl 18401 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
| 17 | 3, 11, 14, 16 | syl3anc 1380 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
| 18 | 1, 17 | eqeltrid 2845 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐷 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 lecple 17222 joincjn 18272 meetcmee 18273 Latclat 18392 Atomscatm 39770 HLchlt 39857 LHypclh 40491 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-lat 18393 df-ats 39774 df-atl 39805 df-cvlat 39829 df-hlat 39858 df-lhyp 40495 |
| This theorem is referenced by: cdleme20k 40826 cdleme20l2 40828 cdleme20l 40829 cdleme20m 40830 |
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