Theorem dalem57 40192

Description: Lemma for dath 40199. Axis of perspectivity point 𝐷 is on the auxiliary line 𝐵. (Contributed by NM, 9-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))))
dalem57.m	⊢ ∧ = (meet‘𝐾)
dalem57.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem57.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem57.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem57.d	⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
dalem57.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem57.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem57.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
dalem57.b1	⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)

Assertion

Ref	Expression
dalem57	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐷 ≤ 𝐵)

Proof of Theorem dalem57

Step	Hyp	Ref	Expression
1		dalem.ph	. . . . . . 7 ⊢ (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		dalem.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
3		dalem.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
4		dalem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5		dalem.ps	. . . . . . 7 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
6		dalem57.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
7		dalem57.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
8		dalem57.y	. . . . . . 7 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
9		dalem57.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
10		dalem57.g	. . . . . . 7 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
11		dalem57.h	. . . . . . 7 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
12		dalem57.i	. . . . . . 7 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
13		dalem57.b1	. . . . . . 7 ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	dalem55 40190	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))
15	1	dalemkelat 40087	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
16	15	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
17	1	dalemkehl 40086	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
18	17	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem23 40159	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem29 40164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
21		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 3, 4	hlatjcl 39830	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
23	18, 19, 20, 22	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
24	1, 3, 4	dalempjqeb 40108	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
25	24	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
26	21, 2, 6	latmle2 18425	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
27	16, 23, 25, 26	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
28	14, 27	eqbrtrrd 5110	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ (𝑃 ∨ 𝑄))
29	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	dalem56 40191	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))
30	1, 3, 4	dalemsjteb 40109	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
31	30	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
32	21, 2, 6	latmle2 18425	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑆 ∨ 𝑇))
33	16, 23, 31, 32	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑆 ∨ 𝑇))
34	29, 33	eqbrtrrd 5110	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ (𝑆 ∨ 𝑇))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	dalem54 40189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)
36	21, 4	atbase 39752	. . . . . . 7 ⊢ (((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴 → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾))
37	35, 36	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾))
38	21, 2, 6	latlem12 18426	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))) → ((((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ (𝑆 ∨ 𝑇)) ↔ ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))))
39	16, 37, 25, 31, 38	syl13anc 1375	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ (𝑆 ∨ 𝑇)) ↔ ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))))
40	28, 34, 39	mpbi2and 713	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)))
41		dalem57.d	. . . 4 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
42	40, 41	breqtrrdi 5128	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐷)
43		hlatl 39823	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
44	18, 43	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat)
45	1, 2, 3, 4, 6, 7, 8, 9, 41	dalemdea 40125	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
46	45	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐷 ∈ 𝐴)
47	2, 4	atcmp 39774	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐷 ↔ ((𝐺 ∨ 𝐻) ∧ 𝐵) = 𝐷))
48	44, 35, 46, 47	syl3anc 1374	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐷 ↔ ((𝐺 ∨ 𝐻) ∧ 𝐵) = 𝐷))
49	42, 48	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) = 𝐷)
50		eqid 2737	. . . . 5 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
51	1, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13	dalem53 40188	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾))
52	21, 50	llnbase 39972	. . . 4 ⊢ (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
53	51, 52	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (Base‘𝐾))
54	21, 2, 6	latmle2 18425	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐵)
55	16, 23, 53, 54	syl3anc 1374	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐵)
56	49, 55	eqbrtrrd 5110	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐷 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221  joincjn 18271  meetcmee 18272  Latclat 18391  Atomscatm 39726  AtLatcal 39727  HLchlt 39813  LLinesclln 39954  LPlanesclpl 39955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-proset 18254  df-poset 18273  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18392  df-clat 18459  df-oposet 39639  df-ol 39641  df-oml 39642  df-covers 39729  df-ats 39730  df-atl 39761  df-cvlat 39785  df-hlat 39814  df-llines 39961  df-lplanes 39962  df-lvols 39963

This theorem is referenced by:  dalem58  40193  dalem60  40195