dalem27 - Mathbox for Norm Megill

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Theorem dalem27 40191

Description: Lemma for dath 40228. 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 40116	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
4	3	3ad2ant1 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
5	2	dalemkehl 40115	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
6	5	3ad2ant1 1139	. . . . 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 40175	. . . . . 6 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
9	8	3ad2ant3 1141	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
10	2	dalempea 40118	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
11	10	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
12		eqid 2739	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
13		dalem.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
14		dalem.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
15	12, 13, 14	hlatjcl 39859	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
16	6, 9, 11, 15	syl3anc 1379	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
17	7	dalemddea 40176	. . . . . 6 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
18	17	3ad2ant3 1141	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
19	2	dalemsea 40121	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
20	19	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
21	12, 13, 14	hlatjcl 39859	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
22	6, 18, 20, 21	syl3anc 1379	. . . 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 18421	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃))
26	4, 16, 22, 25	syl3anc 1379	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃))
27	1, 26	eqbrtrid 5107	. 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 40188	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
32	2, 23, 13, 14, 28, 29	dalemply 40146	. . . . 5 ⊢ (𝜑 → 𝑃 ≤ 𝑌)
33	32	3ad2ant1 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ 𝑌)
34	2, 23, 13, 14, 7, 24, 28, 29, 30, 1	dalem24 40189	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌)
35		nbrne2 5092	. . . . 5 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝑃 ≠ 𝐺)
36	35	necomd 2989	. . . 4 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝐺 ≠ 𝑃)
37	33, 34, 36	syl2anc 590	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≠ 𝑃)
38	23, 13, 14	hlatexch2 39888	. . 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 1391	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃)))
40	27, 39	mpd 15	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 lecple 17218 joincjn 18268 meetcmee 18269 Latclat 18388 Atomscatm 39755 HLchlt 39842 LPlanesclpl 39984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-lat 18389 df-clat 18456 df-oposet 39668 df-ol 39670 df-oml 39671 df-covers 39758 df-ats 39759 df-atl 39790 df-cvlat 39814 df-hlat 39843 df-llines 39990 df-lplanes 39991

This theorem is referenced by: dalem28 40192 dalem32 40196 dalem51 40215 dalem52 40216

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