Theorem dalem17 38551

Description: Lemma for dath 38607. 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 38507	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
3	2	adantr 482	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ (𝑅 ∨ 𝑈))
4	1	dalemkelat 38495	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
6		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 5, 6	dalempjqeb 38516	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
8	1, 6	dalemreb 38512	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
9		eqid 2733	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
10		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
11	9, 10, 5	latlej2 18402	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
12	4, 7, 8, 11	syl3anc 1372	. . . . 5 ⊢ (𝜑 → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
13		dalem17.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
14	12, 13	breqtrrdi 5191	. . . 4 ⊢ (𝜑 → 𝑅 ≤ 𝑌)
15	14	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑅 ≤ 𝑌)
16	1, 5, 6	dalemsjteb 38517	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
17	1, 6	dalemueb 38515	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
18	9, 10, 5	latlej2 18402	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
19	4, 16, 17, 18	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
20		dalem17.z	. . . . . 6 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
21	19, 20	breqtrrdi 5191	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝑍)
22	21	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑈 ≤ 𝑍)
23		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑌 = 𝑍)
24	22, 23	breqtrrd 5177	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑈 ≤ 𝑌)
25		dalem17.o	. . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾)
26	1, 25	dalemyeb 38520	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
27	9, 10, 5	latjle12 18403	. . . . 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 1373	. . . 4 ⊢ (𝜑 → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
29	28	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌) ↔ (𝑅 ∨ 𝑈) ≤ 𝑌))
30	15, 24, 29	mpbi2and 711	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑅 ∨ 𝑈) ≤ 𝑌)
31	1, 6	dalemceb 38509	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
32	1	dalemkehl 38494	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
33	1	dalemrea 38499	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
34	1	dalemuea 38502	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
35	9, 5, 6	hlatjcl 38237	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑅 ∨ 𝑈) ∈ (Base‘𝐾))
36	32, 33, 34, 35	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝑅 ∨ 𝑈) ∈ (Base‘𝐾))
37	9, 10	lattr 18397	. . . 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 1373	. . 3 ⊢ (𝜑 → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
39	38	adantr 482	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝐶 ≤ (𝑅 ∨ 𝑈) ∧ (𝑅 ∨ 𝑈) ≤ 𝑌) → 𝐶 ≤ 𝑌))
40	3, 30, 39	mp2and 698	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  Latclat 18384  Atomscatm 38133  HLchlt 38220  LPlanesclpl 38363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-poset 18266  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-lat 18385  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-lplanes 38370

This theorem is referenced by:  dalem19  38553  dalem25  38569