Theorem dalem56 39722

Description: Lemma for dath 39730. Analogue of dalem55 39721 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 39650	. . . 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 1133	. . 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 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 = 𝑍)
8	7	eqcomd 2735	. . 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 39684	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))))
11		biid 261	. . . 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 261	. . . 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 2729	. . . 4 ⊢ ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃))
18		eqid 2729	. . . 4 ⊢ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)) = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))
19		eqid 2729	. . . 4 ⊢ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)) = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))
20		eqid 2729	. . . 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 39721	. . 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 39618	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
25	24	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
26	1	dalemkehl 39617	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
28	9	dalemccea 39677	. . . . . . . 8 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
29	28	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
30	1	dalempea 39620	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
31	30	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
32		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
33	32, 3, 4	hlatjcl 39360	. . . . . . 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 39678	. . . . . . . 8 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
36	35	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
37	1	dalemsea 39623	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
38	37	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
39	32, 3, 4	hlatjcl 39360	. . . . . . 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 18422	. . . . . 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	eqtrid 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 = ((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)))
44		dalem54.h	. . . . 5 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
45	1	dalemqea 39621	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
46	45	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ 𝐴)
47	32, 3, 4	hlatjcl 39360	. . . . . . 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 39624	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
50	49	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑇 ∈ 𝐴)
51	32, 3, 4	hlatjcl 39360	. . . . . . 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 18422	. . . . . 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	eqtrid 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 = ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄)))
56	43, 55	oveq12d 7405	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) = (((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))))
57	56	oveq1d 7402	. 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 39622	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
61	60	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ 𝐴)
62	32, 3, 4	hlatjcl 39360	. . . . . . . . 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 39625	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
65	64	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑈 ∈ 𝐴)
66	32, 3, 4	hlatjcl 39360	. . . . . . . . 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 18422	. . . . . . . 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	eqtrid 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 = ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅)))
71	56, 70	oveq12d 7405	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))))
72	71, 7	oveq12d 7405	. . . 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 2776	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 = (((((𝑑 ∨ 𝑆) ∧ (𝑐 ∨ 𝑃)) ∨ ((𝑑 ∨ 𝑇) ∧ (𝑐 ∨ 𝑄))) ∨ ((𝑑 ∨ 𝑈) ∧ (𝑐 ∨ 𝑅))) ∧ 𝑍))
74	56, 73	oveq12d 7405	. 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 2774	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  Latclat 18390  Atomscatm 39256  HLchlt 39343  LPlanesclpl 39486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492  df-lplanes 39493  df-lvols 39494

This theorem is referenced by:  dalem57  39723