Theorem dalem57 36867

Description: Lemma for dath 36874. 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 36865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))
15	1	dalemkelat 36762	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
16	15	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
17	1	dalemkehl 36761	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
18	17	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem23 36834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem29 36839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
21		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 3, 4	hlatjcl 36505	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
23	18, 19, 20, 22	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
24	1, 3, 4	dalempjqeb 36783	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
25	24	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
26	21, 2, 6	latmle2 17689	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
27	16, 23, 25, 26	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
28	14, 27	eqbrtrrd 5092	. . . . 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 36866	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))
30	1, 3, 4	dalemsjteb 36784	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
31	30	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
32	21, 2, 6	latmle2 17689	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑆 ∨ 𝑇))
33	16, 23, 31, 32	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑆 ∨ 𝑇))
34	29, 33	eqbrtrrd 5092	. . . . 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 36864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)
36	21, 4	atbase 36427	. . . . . . 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 17690	. . . . . 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 1368	. . . . 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 710	. . . 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 5110	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐷)
43		hlatl 36498	. . . . 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 36800	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
46	45	3ad2ant1 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐷 ∈ 𝐴)
47	2, 4	atcmp 36449	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐷 ↔ ((𝐺 ∨ 𝐻) ∧ 𝐵) = 𝐷))
48	44, 35, 46, 47	syl3anc 1367	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐷 ↔ ((𝐺 ∨ 𝐻) ∧ 𝐵) = 𝐷))
49	42, 48	mpbid 234	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) = 𝐷)
50		eqid 2823	. . . . 5 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
51	1, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13	dalem53 36863	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾))
52	21, 50	llnbase 36647	. . . 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 17689	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐵)
55	16, 23, 53, 54	syl3anc 1367	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≤ 𝐵)
56	49, 55	eqbrtrrd 5092	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐷 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  joincjn 17556  meetcmee 17557  Latclat 17657  Atomscatm 36401  AtLatcal 36402  HLchlt 36488  LLinesclln 36629  LPlanesclpl 36630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-lat 17658  df-clat 17720  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489  df-llines 36636  df-lplanes 36637  df-lvols 36638

This theorem is referenced by:  dalem58  36868  dalem60  36870