dalem27 - Mathbox for Norm Megill

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Theorem dalem27 40736

Description: Lemma for dath 40773. Show that the line 𝐺𝑃 intersects the dummy center of perspectivity 𝑐. (Contributed by NM, 8-Aug-2012.)

**Hypotheses**
Ref	Expression
dalem.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)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
dalem23.m	⊢ ∧ = (meet‘𝐾)
dalem23.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem23.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem23.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem23.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))

**Assertion**
Ref	Expression
dalem27	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃))

**Proof of Theorem dalem27**
Step	Hyp	Ref	Expression
1		dalem23.g	. . 3 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
2		dalem.ph	. . . . . 6 ⊢ (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	2	dalemkelat 40661	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
4	3	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
5	2	dalemkehl 40660	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
6	5	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
7		dalem.ps	. . . . . . 7 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
8	7	dalemccea 40720	. . . . . 6 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
9	8	3ad2ant3 1153	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
10	2	dalempea 40663	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
11	10	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
12		eqid 2761	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
13		dalem.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
14		dalem.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
15	12, 13, 14	hlatjcl 40404	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
16	6, 9, 11, 15	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
17	7	dalemddea 40721	. . . . . 6 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
18	17	3ad2ant3 1153	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
19	2	dalemsea 40666	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
20	19	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
21	12, 13, 14	hlatjcl 40404	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
22	6, 18, 20, 21	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
23		dalem.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
24		dalem23.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
25	12, 23, 24	latmle1 18631	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃))
26	4, 16, 22, 25	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃))
27	1, 26	eqbrtrid 5140	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑐 ∨ 𝑃))
28		dalem23.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
29		dalem23.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
30		dalem23.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
31	2, 23, 13, 14, 7, 24, 28, 29, 30, 1	dalem23 40733	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
32	2, 23, 13, 14, 28, 29	dalemply 40691	. . . . 5 ⊢ (𝜑 → 𝑃 ≤ 𝑌)
33	32	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ 𝑌)
34	2, 23, 13, 14, 7, 24, 28, 29, 30, 1	dalem24 40734	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌)
35		nbrne2 5125	. . . . 5 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝑃 ≠ 𝐺)
36	35	necomd 3011	. . . 4 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝐺 ≠ 𝑃)
37	33, 34, 36	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≠ 𝑃)
38	23, 13, 14	hlatexch2 40433	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝐺 ≠ 𝑃) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃)))
39	6, 31, 9, 11, 37, 38	syl131anc 1410	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃)))
40	27, 39	mpd 16	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃))

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 2956 class class class wbr 5103 ‘cfv 6537 (class class class)co 7418 Basecbs 17380 lecple 17428 joincjn 18478 meetcmee 18479 Latclat 18598 Atomscatm 40300 HLchlt 40387 LPlanesclpl 40529

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-proset 18461 df-poset 18480 df-plt 18495 df-lub 18511 df-glb 18512 df-join 18513 df-meet 18514 df-p0 18590 df-lat 18599 df-clat 18666 df-oposet 40213 df-ol 40215 df-oml 40216 df-covers 40303 df-ats 40304 df-atl 40335 df-cvlat 40359 df-hlat 40388 df-llines 40535 df-lplanes 40536

This theorem is used by: dalem28 40737 dalem32 40741 dalem51 40760 dalem52 40761

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