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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme0aa | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) |
Ref | Expression |
---|---|
cdleme0.l | ⊢ ≤ = (le‘𝐾) |
cdleme0.j | ⊢ ∨ = (join‘𝐾) |
cdleme0.m | ⊢ ∧ = (meet‘𝐾) |
cdleme0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme0.b | ⊢ 𝐵 = (Base‘𝐾) |
Ref | Expression |
---|---|
cdleme0aa | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑈 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme0.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
2 | simp1l 1078 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ HL) | |
3 | hllat 33668 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Lat) |
5 | cdleme0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
6 | cdleme0.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 5, 6 | atbase 33594 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
8 | 7 | 3ad2ant2 1076 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
9 | 5, 6 | atbase 33594 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
10 | 9 | 3ad2ant3 1077 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐵) |
11 | cdleme0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
12 | 5, 11 | latjcl 16874 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
13 | 4, 8, 10, 12 | syl3anc 1318 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
14 | simp1r 1079 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑊 ∈ 𝐻) | |
15 | cdleme0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
16 | 5, 15 | lhpbase 34302 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑊 ∈ 𝐵) |
18 | cdleme0.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
19 | 5, 18 | latmcl 16875 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) |
20 | 4, 13, 17, 19 | syl3anc 1318 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) |
21 | 1, 20 | syl5eqel 2692 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑈 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = 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: cdleme1b 34531 cdleme5 34545 cdleme9 34558 cdleme11g 34570 cdleme11 34575 cdleme35fnpq 34755 cdleme42e 34785 cdlemeg46frv 34831 cdlemg2fv2 34906 cdlemg2m 34910 |
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