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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem9 | Structured version Visualization version GIF version |
Description: Lemma for dath 35525. Since ¬ 𝐶 ≤ 𝑌, the join 𝑌 ∨ 𝐶 forms a 3-dimensional space. (Contributed by NM, 20-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) |
dalem9.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem9.v | ⊢ 𝑉 = (LVols‘𝐾) |
dalem9.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem9.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem9.w | ⊢ 𝑊 = (𝑌 ∨ 𝐶) |
Ref | Expression |
---|---|
dalem9 | ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem9.w | . 2 ⊢ 𝑊 = (𝑌 ∨ 𝐶) | |
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 | 2 | dalemkehl 35412 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ HL) |
4 | 3 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL) |
5 | 2 | dalemyeo 35421 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
6 | 5 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂) |
7 | dalemc.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
8 | dalemc.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
9 | dalemc.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | dalem9.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
11 | dalem9.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
12 | 2, 7, 8, 9, 10, 11 | dalemcea 35449 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
13 | 12 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐶 ∈ 𝐴) |
14 | dalem9.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
15 | 2, 7, 8, 9, 10, 11, 14 | dalem-cly 35460 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ¬ 𝐶 ≤ 𝑌) |
16 | dalem9.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
17 | 7, 8, 9, 10, 16 | lvoli3 35366 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ≤ 𝑌) → (𝑌 ∨ 𝐶) ∈ 𝑉) |
18 | 4, 6, 13, 15, 17 | syl31anc 1480 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝐶) ∈ 𝑉) |
19 | 1, 18 | syl5eqel 2843 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 lecple 16150 joincjn 17145 Atomscatm 35053 HLchlt 35140 LPlanesclpl 35281 LVolsclvol 35282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-lat 17247 df-clat 17309 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 df-lvols 35289 |
This theorem is referenced by: dalem13 35465 dalem14 35466 |
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