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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem8 | Structured version Visualization version GIF version |
Description: Lemma for dath 37487. 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 37419 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊) |
10 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem7 37420 | . . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
11 | 2 | dalemkelat 37375 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | 2, 5 | dalemseb 37393 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
13 | 2, 5 | dalemteb 37394 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
14 | 2, 6 | dalemyeb 37400 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
15 | 2, 5 | dalemceb 37389 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
16 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | 16, 4 | latjcl 17945 | . . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
18 | 11, 14, 15, 17 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
19 | 8, 18 | eqeltrid 2842 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
20 | 16, 3, 4 | latjle12 17956 | . . . . 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 1374 | . . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
22 | 9, 10, 21 | mpbi2and 712 | . . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊) |
23 | 2, 3, 4, 5, 6, 7, 8 | dalem5 37418 | . . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊) |
24 | 2, 4, 5 | dalemsjteb 37397 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
25 | 2, 5 | dalemueb 37395 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
26 | 16, 3, 4 | latjle12 17956 | . . . 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 1374 | . . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
28 | 22, 23, 27 | mpbi2and 712 | . 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊) |
29 | 1, 28 | eqbrtrid 5088 | 1 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 lecple 16809 joincjn 17818 Latclat 17937 Atomscatm 37014 HLchlt 37101 LPlanesclpl 37243 |
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 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-lat 17938 df-clat 18005 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 |
This theorem is referenced by: dalem13 37427 |
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