Theorem dalem56 40352

Description: Lemma for dath 40360. Analogue of dalem55 40351 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 40280	. . . 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 1146	. . 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 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 = 𝑍)
8	7	eqcomd 2768	. . 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 40314	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))))
11		biid 263	. . . 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 263	. . . 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 2762	. . . 4 ⊢ ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃))
18		eqid 2762	. . . 4 ⊢ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)) = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))
19		eqid 2762	. . . 4 ⊢ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)) = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))
20		eqid 2762	. . . 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 40351	. . 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 1390	. 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 40248	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
25	24	3ad2ant1 1146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
26	1	dalemkehl 40247	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
28	9	dalemccea 40307	. . . . . . . 8 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
29	28	3ad2ant3 1148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
30	1	dalempea 40250	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
31	30	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
32		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
33	32, 3, 4	hlatjcl 39991	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
34	27, 29, 31, 33	syl3anc 1390	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
35	9	dalemddea 40308	. . . . . . . 8 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
36	35	3ad2ant3 1148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
37	1	dalemsea 40253	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
38	37	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
39	32, 3, 4	hlatjcl 39991	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
40	27, 36, 38, 39	syl3anc 1390	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
41	32, 13	latmcom 18495	. . . . . 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 1390	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)))
43	23, 42	eqtrid 2809	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)))
44		dalem54.h	. . . . 5 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
45	1	dalemqea 40251	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
46	45	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ 𝐴)
47	32, 3, 4	hlatjcl 39991	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑐 ∨ 𝑄) ∈ (Base‘𝐾))
48	27, 29, 46, 47	syl3anc 1390	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑄) ∈ (Base‘𝐾))
49	1	dalemtea 40254	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
50	49	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑇 ∈ 𝐴)
51	32, 3, 4	hlatjcl 39991	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑑 ∨ 𝑇) ∈ (Base‘𝐾))
52	27, 36, 50, 51	syl3anc 1390	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑇) ∈ (Base‘𝐾))
53	32, 13	latmcom 18495	. . . . . 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 1390	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)))
55	44, 54	eqtrid 2809	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)))
56	43, 55	oveq12d 7414	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) = (((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))))
57	56	oveq1d 7411	. 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 40252	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
61	60	3ad2ant1 1146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ 𝐴)
62	32, 3, 4	hlatjcl 39991	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑐 ∨ 𝑅) ∈ (Base‘𝐾))
63	27, 29, 61, 62	syl3anc 1390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑅) ∈ (Base‘𝐾))
64	1	dalemuea 40255	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
65	64	3ad2ant1 1146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑈 ∈ 𝐴)
66	32, 3, 4	hlatjcl 39991	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑑 ∨ 𝑈) ∈ (Base‘𝐾))
67	27, 36, 65, 66	syl3anc 1390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑈) ∈ (Base‘𝐾))
68	32, 13	latmcom 18495	. . . . . . . 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 1390	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)))
70	59, 69	eqtrid 2809	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)))
71	56, 70	oveq12d 7414	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))))
72	71, 7	oveq12d 7414	. . . 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	eqtrid 2809	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 = (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍))
74	56, 73	oveq12d 7414	. 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 2807	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  lecple 17293  joincjn 18343  meetcmee 18344  Latclat 18463  Atomscatm 39887  HLchlt 39974  LPlanesclpl 40116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-proset 18326  df-poset 18345  df-plt 18360  df-lub 18376  df-glb 18377  df-join 18378  df-meet 18379  df-p0 18455  df-lat 18464  df-clat 18531  df-oposet 39800  df-ol 39802  df-oml 39803  df-covers 39890  df-ats 39891  df-atl 39922  df-cvlat 39946  df-hlat 39975  df-llines 40122  df-lplanes 40123  df-lvols 40124

This theorem is referenced by:  dalem57  40353