Theorem dalem17 36849

Description: Lemma for dath 36905. When planes 𝑌 and 𝑍 are equal, the center of perspectivity 𝐶 is in 𝑌. (Contributed by NM, 1-Aug-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)
dalem17.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem17.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem17.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)

Assertion

Ref	Expression
dalem17	⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)

Proof of Theorem dalem17

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	dalemclrju 36805	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
3	2	adantr 483	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ (𝑅 ∨ 𝑈))
4	1	dalemkelat 36793	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
6		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 5, 6	dalempjqeb 36814	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
8	1, 6	dalemreb 36810	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
9		eqid 2820	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
10		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
11	9, 10, 5	latlej2 17666	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
12	4, 7, 8, 11	syl3anc 1366	. . . . 5 ⊢ (𝜑 → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
13		dalem17.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
14	12, 13	breqtrrdi 5101	. . . 4 ⊢ (𝜑 → 𝑅 ≤ 𝑌)
15	14	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑅 ≤ 𝑌)
16	1, 5, 6	dalemsjteb 36815	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
17	1, 6	dalemueb 36813	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
18	9, 10, 5	latlej2 17666	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
19	4, 16, 17, 18	syl3anc 1366	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
20		dalem17.z	. . . . . 6 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
21	19, 20	breqtrrdi 5101	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝑍)
22	21	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑈 ≤ 𝑍)
23		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑌 = 𝑍)
24	22, 23	breqtrrd 5087	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑈 ≤ 𝑌)
25		dalem17.o	. . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾)
26	1, 25	dalemyeb 36818	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
27	9, 10, 5	latjle12 17667	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
28	4, 8, 17, 26, 27	syl13anc 1367	. . . 4 ⊢ (𝜑 → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
29	28	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
30	15, 24, 29	mpbi2and 710	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑅 ∨ 𝑈) ≤ 𝑌)
31	1, 6	dalemceb 36807	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
32	1	dalemkehl 36792	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
33	1	dalemrea 36797	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
34	1	dalemuea 36800	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
35	9, 5, 6	hlatjcl 36536	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑅 ∨ 𝑈) ∈ (Base‘𝐾))
36	32, 33, 34, 35	syl3anc 1366	. . . 4 ⊢ (𝜑 → (𝑅 ∨ 𝑈) ∈ (Base‘𝐾))
37	9, 10	lattr 17661	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
38	4, 31, 36, 26, 37	syl13anc 1367	. . 3 ⊢ (𝜑 → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
39	38	adantr 483	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
40	3, 30, 39	mp2and 697	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113   class class class wbr 5059  ‘cfv 6348  (class class class)co 7149  Basecbs 16478  lecple 16567  joincjn 17549  Latclat 17650  Atomscatm 36432  HLchlt 36519  LPlanesclpl 36661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-poset 17551  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-lat 17651  df-ats 36436  df-atl 36467  df-cvlat 36491  df-hlat 36520  df-lplanes 36668

This theorem is referenced by:  dalem19  36851  dalem25  36867