Theorem cdleme16 41342

Description: Part of proof of Lemma E in [Crawley] p. 113, conclusion of 3rd paragraph on p. 114. 𝐹 and 𝐺 represent f(s) and f(t) respectively. We show, in their notation, (s ∨ t) ∧ w=(f(s) ∨ f(t)) ∧ w, whether or not u ≤ s ∨ t. (Contributed by NM, 11-Oct-2012.)

Hypotheses

Ref	Expression
cdleme12.l	⊢ ≤ = (le‘𝐾)
cdleme12.j	⊢ ∨ = (join‘𝐾)
cdleme12.m	⊢ ∧ = (meet‘𝐾)
cdleme12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme12.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme12.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme12.g	⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme16	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))

Proof of Theorem cdleme16

Step	Hyp	Ref	Expression
1		simpl11 1267	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl12 1268	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simpl13 1269	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simpl21 1270	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
5		simpl22 1271	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
6		simpl23 1272	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇))
7		simpl3l 1247	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
8		simpl3r 1248	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
9		simpr 490	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → 𝑈 ≤ (𝑆 ∨ 𝑇))
10		cdleme12.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
11		cdleme12.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
12		cdleme12.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
13		cdleme12.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
14		cdleme12.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
15		cdleme12.u	. . . . . 6 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
16		cdleme12.f	. . . . . 6 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
17		cdleme12.g	. . . . . 6 ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
18	10, 11, 12, 13, 14, 15, 16, 17	cdleme11 41327	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝐹 ∨ 𝐺) = (𝑆 ∨ 𝑇))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 18	syl333anc 1429	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝐹 ∨ 𝐺) = (𝑆 ∨ 𝑇))
20	19	eqcomd 2767	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑆 ∨ 𝑇) = (𝐹 ∨ 𝐺))
21	20	oveq1d 7435	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))
22		simpl11 1267	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		simpl12 1268	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
24		simpl13 1269	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
25		simpl21 1270	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
26		simpl22 1271	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
27		simpl23 1272	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇))
28		simpl3l 1247	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
29		simpl3r 1248	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
30		simpr 490	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))
31	10, 11, 12, 13, 14, 15, 16, 17	cdleme16g 41341	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))
32	22, 23, 24, 25, 26, 27, 28, 29, 30, 31	syl333anc 1429	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))
33	21, 32	pm2.61dan 825	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  lecple 17435  joincjn 18485  meetcmee 18486  Atomscatm 40320  HLchlt 40407  LHypclh 41041

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40233  df-ol 40235  df-oml 40236  df-covers 40323  df-ats 40324  df-atl 40355  df-cvlat 40379  df-hlat 40408  df-llines 40555  df-lplanes 40556  df-lvols 40557  df-lines 40558  df-psubsp 40560  df-pmap 40561  df-padd 40853  df-lhyp 41045

This theorem is used by:  cdleme19b  41361  cdleme20d  41369  cdleme22g  41405