cdleme20c - Mathbox for Norm Megill

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Theorem cdleme20c 40571

Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s₂, f(s), t₂, f(t). (Contributed by NM, 15-Nov-2012.)

**Hypotheses**
Ref	Expression
cdleme19.l	⊢ ≤ = (le‘𝐾)
cdleme19.j	⊢ ∨ = (join‘𝐾)
cdleme19.m	⊢ ∧ = (meet‘𝐾)
cdleme19.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme19.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme19.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme19.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme19.g	⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
cdleme19.d	⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
cdleme19.y	⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊)
cdleme20.v	⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊)

**Assertion**
Ref	Expression
cdleme20c	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐷 ∨ 𝑌) = (((𝑅 ∨ 𝑆) ∨ 𝑇) ∧ 𝑊))

**Proof of Theorem cdleme20c**
Step	Hyp	Ref	Expression
1		cdleme19.d	. . 3 ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
2		cdleme19.y	. . 3 ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊)
3	1, 2	oveq12i 7370	. 2 ⊢ (𝐷 ∨ 𝑌) = (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊))
4		simp1l 1198	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
5		simp21l 1291	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
6		simp22l 1293	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ 𝐴)
7		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		cdleme19.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
9		cdleme19.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
10	7, 8, 9	hlatjcl 39627	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
11	4, 5, 6, 10	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
12		simp1r 1199	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻)
13		cdleme19.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
14	7, 13	lhpbase 40258	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
15	12, 14	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ (Base‘𝐾))
16		cdleme19.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
17	16, 8, 9	hlatlej1 39635	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑅 ≤ (𝑅 ∨ 𝑆))
18	4, 5, 6, 17	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑅 ∨ 𝑆))
19		cdleme19.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
20	7, 16, 8, 19, 9	atmod2i1 40121	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑅 ≤ (𝑅 ∨ 𝑆)) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑅) = ((𝑅 ∨ 𝑆) ∧ (𝑊 ∨ 𝑅)))
21	4, 5, 11, 15, 18, 20	syl131anc 1385	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑅) = ((𝑅 ∨ 𝑆) ∧ (𝑊 ∨ 𝑅)))
22		simp21 1207	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
23		eqid 2736	. . . . . . . . . 10 ⊢ (1.‘𝐾) = (1.‘𝐾)
24	16, 8, 23, 9, 13	lhpjat1 40280	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑊 ∨ 𝑅) = (1.‘𝐾))
25	4, 12, 22, 24	syl21anc 837	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑊 ∨ 𝑅) = (1.‘𝐾))
26	25	oveq2d 7374	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∧ (𝑊 ∨ 𝑅)) = ((𝑅 ∨ 𝑆) ∧ (1.‘𝐾)))
27		hlol 39621	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
28	4, 27	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ OL)
29	7, 19, 23	olm11 39487	. . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑆) ∧ (1.‘𝐾)) = (𝑅 ∨ 𝑆))
30	28, 11, 29	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∧ (1.‘𝐾)) = (𝑅 ∨ 𝑆))
31	21, 26, 30	3eqtrrd 2776	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑆) = (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑅))
32	31	oveq1d 7373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∨ 𝑇) = ((((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑅) ∨ 𝑇))
33		simp22r 1294	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑆 ≤ 𝑊)
34		simp3r 1203	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
35		simp3l 1202	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
36		eqid 2736	. . . . . . . 8 ⊢ ((𝑅 ∨ 𝑆) ∧ 𝑊) = ((𝑅 ∨ 𝑆) ∧ 𝑊)
37	16, 8, 19, 9, 13, 36	cdlemeda 40558	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴)
38	4, 12, 6, 33, 5, 34, 35, 37	syl223anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴)
39		simp23 1209	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑇 ∈ 𝐴)
40	8, 9	hlatjass 39630	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑅) ∨ 𝑇) = (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ (𝑅 ∨ 𝑇)))
41	4, 38, 5, 39, 40	syl13anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑅) ∨ 𝑇) = (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ (𝑅 ∨ 𝑇)))
42	32, 41	eqtrd 2771	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∨ 𝑇) = (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ (𝑅 ∨ 𝑇)))
43	42	oveq1d 7373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (((𝑅 ∨ 𝑆) ∨ 𝑇) ∧ 𝑊) = ((((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ (𝑅 ∨ 𝑇)) ∧ 𝑊))
44	7, 8, 9	hlatjcl 39627	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
45	4, 5, 39, 44	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
46	4	hllatd 39624	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
47	7, 16, 19	latmle2 18388	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
48	46, 11, 15, 47	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
49	7, 16, 8, 19, 9	atmod1i1 40117	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴 ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ ((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊)) = ((((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ (𝑅 ∨ 𝑇)) ∧ 𝑊))
50	4, 38, 45, 15, 48, 49	syl131anc 1385	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊)) = ((((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ (𝑅 ∨ 𝑇)) ∧ 𝑊))
51	43, 50	eqtr4d 2774	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (((𝑅 ∨ 𝑆) ∨ 𝑇) ∧ 𝑊) = (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊)))
52	3, 51	eqtr4id 2790	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐷 ∨ 𝑌) = (((𝑅 ∨ 𝑆) ∨ 𝑇) ∧ 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 lecple 17184 joincjn 18234 meetcmee 18235 1.cp1 18345 Latclat 18354 OLcol 39434 Atomscatm 39523 HLchlt 39610 LHypclh 40244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248

This theorem is referenced by: cdleme20d 40572

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