Theorem dalem53 35801

Description: Lemma for dath 35812. The auxliary axis of perspectivity 𝐵 is a line (analogous to the actual axis of perspectivity 𝑋 in dalem15 35754. (Contributed by NM, 8-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))))
dalem53.m	⊢ ∧ = (meet‘𝐾)
dalem53.n	⊢ 𝑁 = (LLines‘𝐾)
dalem53.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem53.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem53.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem53.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem53.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem53.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
dalem53.b1	⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)

Assertion

Ref	Expression
dalem53	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ 𝑁)

Proof of Theorem dalem53

Step	Hyp	Ref	Expression
1		dalem.ph	. . 3 ⊢ (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	. . 3 ⊢ ≤ = (le‘𝐾)
3		dalem.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		dalem.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
5		dalem.ps	. . 3 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
6		dalem53.m	. . 3 ⊢ ∧ = (meet‘𝐾)
7		dalem53.o	. . 3 ⊢ 𝑂 = (LPlanes‘𝐾)
8		dalem53.y	. . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
9		dalem53.z	. . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
10		dalem53.g	. . 3 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
11		dalem53.h	. . 3 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
12		dalem53.i	. . 3 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem51 35799	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌))
14		eqid 2826	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	14, 4	atbase 35365	. . . . 5 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ (Base‘𝐾))
16	15	anim2i 612	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
17	16	3anim1i 1197	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)))
18		biid 253	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))))
19		dalem53.n	. . . 4 ⊢ 𝑁 = (LLines‘𝐾)
20		eqid 2826	. . . 4 ⊢ ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐺 ∨ 𝐻) ∨ 𝐼)
21		dalem53.b1	. . . 4 ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)
22	18, 2, 3, 4, 6, 19, 7, 20, 8, 21	dalem15 35754	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) → 𝐵 ∈ 𝑁)
23	17, 22	syl3anl1 1539	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) → 𝐵 ∈ 𝑁)
24	13, 23	syl 17	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166   ≠ wne 3000   class class class wbr 4874  ‘cfv 6124  (class class class)co 6906  Basecbs 16223  lecple 16313  joincjn 17298  meetcmee 17299  Atomscatm 35339  HLchlt 35426  LLinesclln 35567  LPlanesclpl 35568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-proset 17282  df-poset 17300  df-plt 17312  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-p0 17393  df-lat 17400  df-clat 17462  df-oposet 35252  df-ol 35254  df-oml 35255  df-covers 35342  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427  df-llines 35574  df-lplanes 35575  df-lvols 35576

This theorem is referenced by:  dalem54  35802  dalem55  35803  dalem57  35805  dalem60  35808