Theorem dalem29 40575

Description: Lemma for dath 40610. Analogue of dalem23 40570 for 𝐻. (Contributed by NM, 2-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))))
dalem29.m	⊢ ∧ = (meet‘𝐾)
dalem29.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem29.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem29.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem29.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))

Assertion

Ref	Expression
dalem29	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)

Proof of Theorem dalem29

Step	Hyp	Ref	Expression
1		dalem.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		dalem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dalem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		dalem29.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
6		dalem29.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
7	1, 2, 3, 4, 5, 6	dalemrot 40531	. . 3 ⊢ (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))))
8	7	3ad2ant1 1151	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))))
9	1, 2, 3, 4, 5, 6	dalemrotyz 40532	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑇 ∨ 𝑈) ∨ 𝑆))
10	9	3adant3 1150	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑇 ∨ 𝑈) ∨ 𝑆))
11		dalem.ps	. . . 4 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
12	1, 2, 3, 4, 11, 5	dalemrotps 40565	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
13	12	3adant2 1149	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
14		biid 264	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))))
15		biid 264	. . 3 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
16		dalem29.m	. . 3 ⊢ ∧ = (meet‘𝐾)
17		dalem29.o	. . 3 ⊢ 𝑂 = (LPlanes‘𝐾)
18		eqid 2760	. . 3 ⊢ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑃)
19		eqid 2760	. . 3 ⊢ ((𝑇 ∨ 𝑈) ∨ 𝑆) = ((𝑇 ∨ 𝑈) ∨ 𝑆)
20		dalem29.h	. . 3 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
21	14, 2, 3, 4, 15, 16, 17, 18, 19, 20	dalem23 40570	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))) ∧ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑇 ∨ 𝑈) ∨ 𝑆) ∧ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) → 𝐻 ∈ 𝐴)
22	8, 10, 13, 21	syl3anc 1398	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  lecple 17350  joincjn 18400  meetcmee 18401  Atomscatm 40137  HLchlt 40224  LPlanesclpl 40366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-proset 18383  df-poset 18402  df-plt 18417  df-lub 18433  df-glb 18434  df-join 18435  df-meet 18436  df-p0 18512  df-lat 18521  df-clat 18588  df-oposet 40050  df-ol 40052  df-oml 40053  df-covers 40140  df-ats 40141  df-atl 40172  df-cvlat 40196  df-hlat 40225  df-llines 40372  df-lplanes 40373

This theorem is used by:  dalem34  40580  dalem38  40584  dalem39  40585  dalem41  40587  dalem42  40588  dalem43  40589  dalem44  40590  dalem45  40591  dalem51  40597  dalem52  40598  dalem54  40600  dalem55  40601  dalem57  40603  dalem58  40604  dalem59  40605