Theorem dalem10 40165

Description: Lemma for dath 40228. Atom 𝐷 belongs to the axis of perspectivity 𝑋. (Contributed by NM, 19-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)
dalem10.m	⊢ ∧ = (meet‘𝐾)
dalem10.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem10.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem10.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem10.x	⊢ 𝑋 = (𝑌 ∧ 𝑍)
dalem10.d	⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))

Assertion

Ref	Expression
dalem10	⊢ (𝜑 → 𝐷 ≤ 𝑋)

Proof of Theorem dalem10

Step	Hyp	Ref	Expression
1		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)))))
2	1	dalemkelat 40116	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
3		dalemc.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		dalemc.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 3, 4	dalempjqeb 40137	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
6	1, 4	dalemreb 40133	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
7		eqid 2739	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		dalemc.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
9	7, 8, 3	latlej1 18405	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
10	2, 5, 6, 9	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
11	1, 3, 4	dalemsjteb 40138	. . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
12	1, 4	dalemueb 40136	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
13	7, 8, 3	latlej1 18405	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
14	2, 11, 12, 13	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
15		dalem10.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
16		dalem10.o	. . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾)
17	1, 16	dalemyeb 40141	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
18	15, 17	eqeltrrid 2844	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾))
19		dalem10.z	. . . . 5 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
20	1	dalemzeo 40125	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
21	7, 16	lplnbase 40026	. . . . . 6 ⊢ (𝑍 ∈ 𝑂 → 𝑍 ∈ (Base‘𝐾))
22	20, 21	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐾))
23	19, 22	eqeltrrid 2844	. . . 4 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))
24		dalem10.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
25	7, 8, 24	latmlem12 18428	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾)) ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈))))
26	2, 5, 18, 11, 23, 25	syl122anc 1387	. . 3 ⊢ (𝜑 → (((𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈))))
27	10, 14, 26	mp2and 705	. 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
28		dalem10.d	. 2 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
29		dalem10.x	. . 3 ⊢ 𝑋 = (𝑌 ∧ 𝑍)
30	15, 19	oveq12i 7368	. . 3 ⊢ (𝑌 ∧ 𝑍) = (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈))
31	29, 30	eqtri 2762	. 2 ⊢ 𝑋 = (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈))
32	27, 28, 31	3brtr4g 5106	1 ⊢ (𝜑 → 𝐷 ≤ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   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-poset 18270  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-lat 18389  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-lplanes 39991

This theorem is referenced by:  dalem11  40166  dalem16  40171  dalem54  40218