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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem8 | Structured version Visualization version GIF version |
Description: Lemma for dath 34040. Plane 𝑍 belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-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) |
dalem6.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem6.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem6.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem6.w | ⊢ 𝑊 = (𝑌 ∨ 𝐶) |
Ref | Expression |
---|---|
dalem8 | ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem6.z | . 2 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
2 | dalema.ph | . . . . 5 ⊢ (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 | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
4 | dalemc.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
5 | dalemc.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dalem6.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
7 | dalem6.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
8 | dalem6.w | . . . . 5 ⊢ 𝑊 = (𝑌 ∨ 𝐶) | |
9 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem6 33972 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊) |
10 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem7 33973 | . . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
11 | 2 | dalemkelat 33928 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | 2, 5 | dalemseb 33946 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
13 | 2, 5 | dalemteb 33947 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
14 | 2, 6 | dalemyeb 33953 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
15 | 2, 5 | dalemceb 33942 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
16 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | 16, 4 | latjcl 16874 | . . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
18 | 11, 14, 15, 17 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
19 | 8, 18 | syl5eqel 2692 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
20 | 16, 3, 4 | latjle12 16885 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
21 | 11, 12, 13, 19, 20 | syl13anc 1320 | . . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
22 | 9, 10, 21 | mpbi2and 958 | . . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊) |
23 | 2, 3, 4, 5, 6, 7, 8 | dalem5 33971 | . . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊) |
24 | 2, 4, 5 | dalemsjteb 33950 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
25 | 2, 5 | dalemueb 33948 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
26 | 16, 3, 4 | latjle12 16885 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
27 | 11, 24, 25, 19, 26 | syl13anc 1320 | . . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
28 | 22, 23, 27 | mpbi2and 958 | . 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊) |
29 | 1, 28 | syl5eqbr 4618 | 1 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 Latclat 16868 Atomscatm 33568 HLchlt 33655 LPlanesclpl 33796 |
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-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 |
This theorem is referenced by: dalem13 33980 |
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