dalem13 - Mathbox for Norm Megill

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Theorem dalem13 39665

Description: Lemma for dalem14 39666. (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 39612	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
4	1	dalemyeo 39621	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂)
6	1	dalemzeo 39622	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
7	6	adantr 480	. 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 2730	. . 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 39661	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾))
17	1	dalemkelat 39613	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 11	dalemyeb 39638	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
19	1, 10	dalemceb 39627	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
20		eqid 2730	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 8, 9	latlej1 18413	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶))
22	17, 18, 19, 21	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶))
23	22, 15	breqtrrdi 5151	. . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
24	23	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊)
25	1, 8, 9, 10, 11, 13, 14, 15	dalem8 39659	. . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊)
26	25	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊)
27		simpr 484	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
28	8, 9, 11, 12	2lplnj 39609	. 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 1395	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 Basecbs 17185 lecple 17233 joincjn 18278 Latclat 18396 Atomscatm 39251 HLchlt 39338 LPlanesclpl 39481 LVolsclvol 39482

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-proset 18261 df-poset 18280 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-clat 18464 df-oposet 39164 df-ol 39166 df-oml 39167 df-covers 39254 df-ats 39255 df-atl 39286 df-cvlat 39310 df-hlat 39339 df-llines 39487 df-lplanes 39488 df-lvols 39489

This theorem is referenced by: dalem14 39666

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