Theorem dalem56 37428

Description: Lemma for dath 37436. Analogue of dalem55 37427 for line 𝑆𝑇. (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))))
dalem54.m	⊢ ∧ = (meet‘𝐾)
dalem54.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem54.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem54.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem54.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem54.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem54.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
dalem54.b1	⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)

Assertion

Ref	Expression
dalem56	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Proof of Theorem dalem56

Step	Hyp	Ref	Expression
1		dalem.ph	. . . . 5 ⊢ (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	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		dalem.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		dalem.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	dalemswapyz 37356	. . . 4 ⊢ (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (𝐶 ≤ (𝑆 ∨ 𝑃) ∧ 𝐶 ≤ (𝑇 ∨ 𝑄) ∧ 𝐶 ≤ (𝑈 ∨ 𝑅)))))
6	5	3ad2ant1 1135	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (𝐶 ≤ (𝑆 ∨ 𝑃) ∧ 𝐶 ≤ (𝑇 ∨ 𝑄) ∧ 𝐶 ≤ (𝑈 ∨ 𝑅)))))
7		simp2 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 = 𝑍)
8	7	eqcomd 2742	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑍 = 𝑌)
9		dalem.ps	. . . 4 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
10	1, 2, 3, 4, 9	dalemswapyzps 37390	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))))
11		biid 264	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (𝐶 ≤ (𝑆 ∨ 𝑃) ∧ 𝐶 ≤ (𝑇 ∨ 𝑄) ∧ 𝐶 ≤ (𝑈 ∨ 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (𝐶 ≤ (𝑆 ∨ 𝑃) ∧ 𝐶 ≤ (𝑇 ∨ 𝑄) ∧ 𝐶 ≤ (𝑈 ∨ 𝑅)))))
12		biid 264	. . . 4 ⊢ (((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))) ↔ ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))))
13		dalem54.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
14		dalem54.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
15		dalem54.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
16		dalem54.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
17		eqid 2736	. . . 4 ⊢ ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃))
18		eqid 2736	. . . 4 ⊢ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)) = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))
19		eqid 2736	. . . 4 ⊢ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)) = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))
20		eqid 2736	. . . 4 ⊢ (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍) = (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍)
21	11, 2, 3, 4, 12, 13, 14, 15, 16, 17, 18, 19, 20	dalem55 37427	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (𝐶 ≤ (𝑆 ∨ 𝑃) ∧ 𝐶 ≤ (𝑇 ∨ 𝑄) ∧ 𝐶 ≤ (𝑈 ∨ 𝑅)))) ∧ 𝑍 = 𝑌 ∧ ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐)))) → ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∧ (𝑆 ∨ 𝑇)) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∧ (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍)))
22	6, 8, 10, 21	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∧ (𝑆 ∨ 𝑇)) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∧ (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍)))
23		dalem54.g	. . . . 5 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
24	1	dalemkelat 37324	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
25	24	3ad2ant1 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
26	1	dalemkehl 37323	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	3ad2ant1 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
28	9	dalemccea 37383	. . . . . . . 8 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
29	28	3ad2ant3 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
30	1	dalempea 37326	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
31	30	3ad2ant1 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
32		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
33	32, 3, 4	hlatjcl 37067	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
34	27, 29, 31, 33	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
35	9	dalemddea 37384	. . . . . . . 8 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
36	35	3ad2ant3 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
37	1	dalemsea 37329	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
38	37	3ad2ant1 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
39	32, 3, 4	hlatjcl 37067	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
40	27, 36, 38, 39	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
41	32, 13	latmcom 17923	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)))
42	25, 34, 40, 41	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)))
43	23, 42	syl5eq 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)))
44		dalem54.h	. . . . 5 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
45	1	dalemqea 37327	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
46	45	3ad2ant1 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ 𝐴)
47	32, 3, 4	hlatjcl 37067	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑐 ∨ 𝑄) ∈ (Base‘𝐾))
48	27, 29, 46, 47	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑄) ∈ (Base‘𝐾))
49	1	dalemtea 37330	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
50	49	3ad2ant1 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑇 ∈ 𝐴)
51	32, 3, 4	hlatjcl 37067	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑑 ∨ 𝑇) ∈ (Base‘𝐾))
52	27, 36, 50, 51	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑇) ∈ (Base‘𝐾))
53	32, 13	latmcom 17923	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)))
54	25, 48, 52, 53	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)))
55	44, 54	syl5eq 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)))
56	43, 55	oveq12d 7209	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) = (((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))))
57	56	oveq1d 7206	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∧ (𝑆 ∨ 𝑇)))
58		dalem54.b1	. . . 4 ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)
59		dalem54.i	. . . . . . 7 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
60	1	dalemrea 37328	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
61	60	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ 𝐴)
62	32, 3, 4	hlatjcl 37067	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑐 ∨ 𝑅) ∈ (Base‘𝐾))
63	27, 29, 61, 62	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑅) ∈ (Base‘𝐾))
64	1	dalemuea 37331	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
65	64	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑈 ∈ 𝐴)
66	32, 3, 4	hlatjcl 37067	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑑 ∨ 𝑈) ∈ (Base‘𝐾))
67	27, 36, 65, 66	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑈) ∈ (Base‘𝐾))
68	32, 13	latmcom 17923	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑅) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑈) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)))
69	25, 63, 67, 68	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)))
70	59, 69	syl5eq 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)))
71	56, 70	oveq12d 7209	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))))
72	71, 7	oveq12d 7209	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌) = (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍))
73	58, 72	syl5eq 2783	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 = (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍))
74	56, 73	oveq12d 7209	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∧ (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍)))
75	22, 57, 74	3eqtr4d 2781	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  lecple 16756  joincjn 17772  meetcmee 17773  Latclat 17891  Atomscatm 36963  HLchlt 37050  LPlanesclpl 37192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-proset 17756  df-poset 17774  df-plt 17790  df-lub 17806  df-glb 17807  df-join 17808  df-meet 17809  df-p0 17885  df-lat 17892  df-clat 17959  df-oposet 36876  df-ol 36878  df-oml 36879  df-covers 36966  df-ats 36967  df-atl 36998  df-cvlat 37022  df-hlat 37051  df-llines 37198  df-lplanes 37199  df-lvols 37200

This theorem is referenced by:  dalem57  37429