dalem13 - Mathbox for Norm Megill

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Theorem dalem13 35484

Description: Lemma for dalem14 35485. (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 35431	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
4	1	dalemyeo 35440	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂)
6	1	dalemzeo 35441	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
7	6	adantr 466	. 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 2771	. . 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 35480	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾))
17	1	dalemkelat 35432	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 11	dalemyeb 35457	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
19	1, 10	dalemceb 35446	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
20		eqid 2771	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 8, 9	latlej1 17268	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶))
22	17, 18, 19, 21	syl3anc 1476	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶))
23	22, 15	syl6breqr 4828	. . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
24	23	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊)
25	1, 8, 9, 10, 11, 13, 14, 15	dalem8 35478	. . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊)
26	25	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊)
27		simpr 471	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
28	8, 9, 11, 12	2lplnj 35428	. 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 1499	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 lecple 16156 joincjn 17152 Latclat 17253 Atomscatm 35072 HLchlt 35159 LPlanesclpl 35300 LVolsclvol 35301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-lat 17254 df-clat 17316 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308

This theorem is referenced by: dalem14 35485

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