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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemdea | Structured version Visualization version GIF version |
Description: Lemma for dath 35022. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
Ref | Expression |
---|---|
dalema.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) |
dalemc.l | ⊢ ≤ = (le‘𝐾) |
dalemc.j | ⊢ ∨ = (join‘𝐾) |
dalemc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalemdea.m | ⊢ ∧ = (meet‘𝐾) |
dalemdea.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalemdea.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalemdea.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalemdea.d | ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) |
Ref | Expression |
---|---|
dalemdea | ⊢ (𝜑 → 𝐷 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalemdea.d | . 2 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) | |
2 | dalema.ph | . . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) | |
3 | dalemc.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | dalemc.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | dalemc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dalemdea.o | . . . 4 ⊢ 𝑂 = (LPlanes‘𝐾) | |
7 | dalemdea.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
8 | 2, 3, 4, 5, 6, 7 | dalem2 34947 | . . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ 𝑂) |
9 | 2 | dalemkehl 34909 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ HL) |
10 | 2 | dalempea 34912 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
11 | 2 | dalemqea 34913 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
12 | 2 | dalemrea 34914 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
13 | 2 | dalemyeo 34918 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
14 | 4, 5, 6, 7 | lplnri1 34839 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑌 ∈ 𝑂) → 𝑃 ≠ 𝑄) |
15 | 9, 10, 11, 12, 13, 14 | syl131anc 1339 | . . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑄) |
16 | eqid 2622 | . . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
17 | 4, 5, 16 | llni2 34798 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ (LLines‘𝐾)) |
18 | 9, 10, 11, 15, 17 | syl31anc 1329 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (LLines‘𝐾)) |
19 | 2 | dalemsea 34915 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
20 | 2 | dalemtea 34916 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
21 | 2 | dalemuea 34917 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
22 | 2 | dalemzeo 34919 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
23 | dalemdea.z | . . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
24 | 4, 5, 6, 23 | lplnri1 34839 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ 𝑍 ∈ 𝑂) → 𝑆 ≠ 𝑇) |
25 | 9, 19, 20, 21, 22, 24 | syl131anc 1339 | . . . . 5 ⊢ (𝜑 → 𝑆 ≠ 𝑇) |
26 | 4, 5, 16 | llni2 34798 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑆 ≠ 𝑇) → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) |
27 | 9, 19, 20, 25, 26 | syl31anc 1329 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) |
28 | dalemdea.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
29 | 4, 28, 5, 16, 6 | 2llnmj 34846 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (LLines‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ 𝑂)) |
30 | 9, 18, 27, 29 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ 𝑂)) |
31 | 8, 30 | mpbird 247 | . 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴) |
32 | 1, 31 | syl5eqel 2705 | 1 ⊢ (𝜑 → 𝐷 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 joincjn 16944 meetcmee 16945 Atomscatm 34550 HLchlt 34637 LLinesclln 34777 LPlanesclpl 34778 |
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-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 |
This theorem is referenced by: dalemeea 34949 dalem3 34950 dalem16 34965 dalem52 35010 dalem57 35015 dalem60 35018 |
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