dalem8 - Mathbox for Norm Megill

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Theorem dalem8 40177

Description: Lemma for dath 40243. Plane 𝑍 belongs to the 3-dimensional space. (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)
dalem6.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem6.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem6.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem6.w	⊢ 𝑊 = (𝑌 ∨ 𝐶)

**Assertion**
Ref	Expression
dalem8	⊢ (𝜑 → 𝑍 ≤ 𝑊)

**Proof of Theorem dalem8**
Step	Hyp	Ref	Expression
1		dalem6.z	. 2 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
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		dalemc.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		dalemc.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		dalemc.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		dalem6.o	. . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾)
7		dalem6.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
8		dalem6.w	. . . . 5 ⊢ 𝑊 = (𝑌 ∨ 𝐶)
9	2, 3, 4, 5, 6, 7, 1, 8	dalem6 40175	. . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊)
10	2, 3, 4, 5, 6, 7, 1, 8	dalem7 40176	. . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
11	2	dalemkelat 40131	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
12	2, 5	dalemseb 40149	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
13	2, 5	dalemteb 40150	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
14	2, 6	dalemyeb 40156	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
15	2, 5	dalemceb 40145	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
16		eqid 2741	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
17	16, 4	latjcl 18400	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
18	11, 14, 15, 17	syl3anc 1380	. . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
19	8, 18	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
20	16, 3, 4	latjle12 18411	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊))
21	11, 12, 13, 19, 20	syl13anc 1381	. . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊))
22	9, 10, 21	mpbi2and 719	. . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊)
23	2, 3, 4, 5, 6, 7, 8	dalem5 40174	. . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
24	2, 4, 5	dalemsjteb 40153	. . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
25	2, 5	dalemueb 40151	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
26	16, 3, 4	latjle12 18411	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊))
27	11, 24, 25, 19, 26	syl13anc 1381	. . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊))
28	22, 23, 27	mpbi2and 719	. 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)
29	1, 28	eqbrtrid 5110	1 ⊢ (𝜑 → 𝑍 ≤ 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 lecple 17222 joincjn 18272 Latclat 18392 Atomscatm 39770 HLchlt 39857 LPlanesclpl 39999

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-lat 18393 df-clat 18460 df-oposet 39683 df-ol 39685 df-oml 39686 df-covers 39773 df-ats 39774 df-atl 39805 df-cvlat 39829 df-hlat 39858 df-llines 40005 df-lplanes 40006

This theorem is referenced by: dalem13 40183

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