Theorem dalem37 40680

Description: Lemma for dath 40707. Analogue of dalem28 40671 for 𝐼. (Contributed by NM, 4-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))))
dalem34.m	⊢ ∧ = (meet‘𝐾)
dalem34.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem34.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem34.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem34.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))

Assertion

Ref	Expression
dalem37	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))

Proof of Theorem dalem37

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		dalem34.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
6		dalem34.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
7	1, 2, 3, 4, 5, 6	dalemrot 40628	. . 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 40629	. . 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 40662	. . 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		dalem34.m	. . 3 ⊢ ∧ = (meet‘𝐾)
17		dalem34.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		dalem34.i	. . 3 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
21	14, 2, 3, 4, 15, 16, 17, 18, 19, 20	dalem33 40676	. 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)))) → 𝑅 ≤ (𝐼 ∨ 𝑐))
22	8, 10, 13, 21	syl3anc 1398	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))

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 6528  (class class class)co 7409  Basecbs 17334  lecple 17382  joincjn 18432  meetcmee 18433  Atomscatm 40234  HLchlt 40321  LPlanesclpl 40463

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 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

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 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7366  df-ov 7412  df-oprab 7413  df-proset 18415  df-poset 18434  df-plt 18449  df-lub 18465  df-glb 18466  df-join 18467  df-meet 18468  df-p0 18544  df-lat 18553  df-clat 18620  df-oposet 40147  df-ol 40149  df-oml 40150  df-covers 40237  df-ats 40238  df-atl 40269  df-cvlat 40293  df-hlat 40322  df-llines 40469  df-lplanes 40470

This theorem is used by:  dalem38  40681  dalem44  40687