dalem8 - Mathbox for Norm Megill

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Theorem dalem8 35477

Description: Lemma for dath 35543. Plane 𝑍 belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-2012.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dalem8	⊢ (𝜑 → 𝑍 ≤ 𝑊)

**Proof of Theorem dalem8**
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 35475	. . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊)
10	2, 3, 4, 5, 6, 7, 1, 8	dalem7 35476	. . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
11	2	dalemkelat 35431	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
12	2, 5	dalemseb 35449	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
13	2, 5	dalemteb 35450	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
14	2, 6	dalemyeb 35456	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
15	2, 5	dalemceb 35445	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
16		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
17	16, 4	latjcl 17272	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
18	11, 14, 15, 17	syl3anc 1477	. . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
19	8, 18	syl5eqel 2843	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
20	16, 3, 4	latjle12 17283	. . . . 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 1479	. . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊))
22	9, 10, 21	mpbi2and 994	. . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊)
23	2, 3, 4, 5, 6, 7, 8	dalem5 35474	. . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
24	2, 4, 5	dalemsjteb 35453	. . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
25	2, 5	dalemueb 35451	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
26	16, 3, 4	latjle12 17283	. . . 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 1479	. . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊))
28	22, 23, 27	mpbi2and 994	. 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)
29	1, 28	syl5eqbr 4839	1 ⊢ (𝜑 → 𝑍 ≤ 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 lecple 16170 joincjn 17165 Latclat 17266 Atomscatm 35071 HLchlt 35158 LPlanesclpl 35299

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 7115

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 6775 df-ov 6817 df-oprab 6818 df-preset 17149 df-poset 17167 df-plt 17179 df-lub 17195 df-glb 17196 df-join 17197 df-meet 17198 df-p0 17260 df-lat 17267 df-clat 17329 df-oposet 34984 df-ol 34986 df-oml 34987 df-covers 35074 df-ats 35075 df-atl 35106 df-cvlat 35130 df-hlat 35159 df-llines 35305 df-lplanes 35306

This theorem is referenced by: dalem13 35483

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