dalem13 - Mathbox for Norm Megill

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Theorem dalem13 35456

Description: Lemma for dalem14 35457. (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)
dalem13.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem13.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem13.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem13.w	⊢ 𝑊 = (𝑌 ∨ 𝐶)

**Assertion**
Ref	Expression
dalem13	⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊)

**Proof of Theorem dalem13**
Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 ⊢ (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)))))
2	1	dalemkehl 35403	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	adantr 468	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
4	1	dalemyeo 35412	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	adantr 468	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂)
6	1	dalemzeo 35413	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
7	6	adantr 468	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑂)
8		dalemc.l	. . 3 ⊢ ≤ = (le‘𝐾)
9		dalemc.j	. . 3 ⊢ ∨ = (join‘𝐾)
10		dalemc.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
11		dalem13.o	. . 3 ⊢ 𝑂 = (LPlanes‘𝐾)
12		eqid 2806	. . 3 ⊢ (LVols‘𝐾) = (LVols‘𝐾)
13		dalem13.y	. . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
14		dalem13.z	. . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
15		dalem13.w	. . 3 ⊢ 𝑊 = (𝑌 ∨ 𝐶)
16	1, 8, 9, 10, 11, 12, 13, 14, 15	dalem9 35452	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾))
17	1	dalemkelat 35404	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 11	dalemyeb 35429	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
19	1, 10	dalemceb 35418	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
20		eqid 2806	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 8, 9	latlej1 17265	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶))
22	17, 18, 19, 21	syl3anc 1483	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶))
23	22, 15	syl6breqr 4886	. . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
24	23	adantr 468	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊)
25	1, 8, 9, 10, 11, 13, 14, 15	dalem8 35450	. . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊)
26	25	adantr 468	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊)
27		simpr 473	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
28	8, 9, 11, 12	2lplnj 35400	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ∧ 𝑊 ∈ (LVols‘𝐾)) ∧ (𝑌 ≤ 𝑊 ∧ 𝑍 ≤ 𝑊 ∧ 𝑌 ≠ 𝑍)) → (𝑌 ∨ 𝑍) = 𝑊)
29	3, 5, 7, 16, 24, 26, 27, 28	syl133anc 1505	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 197 ∧ wa 384 ∧ w3a 1100 = wceq 1637 ∈ wcel 2156 ≠ wne 2978 class class class wbr 4844 ‘cfv 6101 (class class class)co 6874 Basecbs 16068 lecple 16160 joincjn 17149 Latclat 17250 Atomscatm 35043 HLchlt 35130 LPlanesclpl 35272 LVolsclvol 35273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7179

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-ral 3101 df-rex 3102 df-reu 3103 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-op 4377 df-uni 4631 df-iun 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5219 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6835 df-ov 6877 df-oprab 6878 df-proset 17133 df-poset 17151 df-plt 17163 df-lub 17179 df-glb 17180 df-join 17181 df-meet 17182 df-p0 17244 df-lat 17251 df-clat 17313 df-oposet 34956 df-ol 34958 df-oml 34959 df-covers 35046 df-ats 35047 df-atl 35078 df-cvlat 35102 df-hlat 35131 df-llines 35278 df-lplanes 35279 df-lvols 35280

This theorem is referenced by: dalem14 35457

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