Theorem dalem17 33984

Description: Lemma for dath 34040. 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 33940	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ (𝑅 ∨ 𝑈))
4	1	dalemkelat 33928	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
6		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 5, 6	dalempjqeb 33949	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
8	1, 6	dalemreb 33945	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
9		eqid 2610	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
10		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
11	9, 10, 5	latlej2 16884	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
12	4, 7, 8, 11	syl3anc 1318	. . . . 5 ⊢ (𝜑 → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
13		dalem17.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
14	12, 13	syl6breqr 4625	. . . 4 ⊢ (𝜑 → 𝑅 ≤ 𝑌)
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑅 ≤ 𝑌)
16	1, 5, 6	dalemsjteb 33950	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
17	1, 6	dalemueb 33948	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
18	9, 10, 5	latlej2 16884	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
19	4, 16, 17, 18	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
20		dalem17.z	. . . . . 6 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
21	19, 20	syl6breqr 4625	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝑍)
22	21	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑈 ≤ 𝑍)
23		simpr 476	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑌 = 𝑍)
24	22, 23	breqtrrd 4611	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑈 ≤ 𝑌)
25		dalem17.o	. . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾)
26	1, 25	dalemyeb 33953	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
27	9, 10, 5	latjle12 16885	. . . . 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 1320	. . . 4 ⊢ (𝜑 → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
29	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
30	15, 24, 29	mpbi2and 958	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑅 ∨ 𝑈) ≤ 𝑌)
31	1, 6	dalemceb 33942	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
32	1	dalemkehl 33927	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
33	1	dalemrea 33932	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
34	1	dalemuea 33935	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
35	9, 5, 6	hlatjcl 33671	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑅 ∨ 𝑈) ∈ (Base‘𝐾))
36	32, 33, 34, 35	syl3anc 1318	. . . 4 ⊢ (𝜑 → (𝑅 ∨ 𝑈) ∈ (Base‘𝐾))
37	9, 10	lattr 16879	. . . 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 1320	. . 3 ⊢ (𝜑 → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
39	38	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
40	3, 30, 39	mp2and 711	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655  LPlanesclpl 33796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-lplanes 33803

This theorem is referenced by:  dalem19  33986  dalem25  34002