dalem13 - Mathbox for Norm Megill

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Theorem dalem13 40550

Description: Lemma for dalem14 40551. (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 40497	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
4	1	dalemyeo 40506	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂)
6	1	dalemzeo 40507	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
7	6	adantr 486	. 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 2760	. . 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 40546	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾))
17	1	dalemkelat 40498	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 11	dalemyeb 40523	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
19	1, 10	dalemceb 40512	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
20		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 8, 9	latlej1 18537	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶))
22	17, 18, 19, 21	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶))
23	22, 15	breqtrrdi 5147	. . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
24	23	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊)
25	1, 8, 9, 10, 11, 13, 14, 15	dalem8 40544	. . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊)
26	25	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊)
27		simpr 490	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
28	8, 9, 11, 12	2lplnj 40494	. 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 1420	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 lecple 17350 joincjn 18400 Latclat 18520 Atomscatm 40137 HLchlt 40224 LPlanesclpl 40366 LVolsclvol 40367

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-proset 18383 df-poset 18402 df-plt 18417 df-lub 18433 df-glb 18434 df-join 18435 df-meet 18436 df-p0 18512 df-lat 18521 df-clat 18588 df-oposet 40050 df-ol 40052 df-oml 40053 df-covers 40140 df-ats 40141 df-atl 40172 df-cvlat 40196 df-hlat 40225 df-llines 40372 df-lplanes 40373 df-lvols 40374

This theorem is used by: dalem14 40551

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