dalem9 - Mathbox for Norm Megill

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Theorem dalem9 33976

Description: Lemma for dath 34040. Since ¬ 𝐶 ≤ 𝑌, the join 𝑌 ∨ 𝐶 forms a 3-dimensional space. (Contributed by NM, 20-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)
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)

**Assertion**
Ref	Expression
dalem9	⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ 𝑉)

**Proof of Theorem dalem9**
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 33927	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
5	2	dalemyeo 33936	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
6	5	adantr 480	. . 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 33964	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐶 ∈ 𝐴)
14		dalem9.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
15	2, 7, 8, 9, 10, 11, 14	dalem-cly 33975	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ¬ 𝐶 ≤ 𝑌)
16		dalem9.v	. . . 4 ⊢ 𝑉 = (LVols‘𝐾)
17	7, 8, 9, 10, 16	lvoli3 33881	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ≤ 𝑌) → (𝑌 ∨ 𝐶) ∈ 𝑉)
18	4, 6, 13, 15, 17	syl31anc 1321	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝐶) ∈ 𝑉)
19	1, 18	syl5eqel 2692	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ 𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 Atomscatm 33568 HLchlt 33655 LPlanesclpl 33796 LVolsclvol 33797

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 df-lvols 33804

This theorem is referenced by: dalem13 33980 dalem14 33981

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