Theorem cdleme12 40731

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. 𝐹 and 𝐺 represent f(s) and f(t) respectively. (Contributed by NM, 16-Jun-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
cdleme12	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝑆 ∨ 𝐹) ∧ (𝑇 ∨ 𝐺)) = 𝑈)

Proof of Theorem cdleme12

Step	Hyp	Ref	Expression
1		simp1 1137	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp21l 1292	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑃 ∈ 𝐴)
3		simp22 1209	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑄 ∈ 𝐴)
4		simp31 1211	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
5		cdleme12.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
6		cdleme12.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
7		cdleme12.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
8		cdleme12.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdleme12.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdleme12.u	. . . . . 6 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
11		cdleme12.f	. . . . . 6 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
12	5, 6, 7, 8, 9, 10, 11	cdleme1 40687	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝑆 ∨ 𝐹) = (𝑆 ∨ 𝑈))
13	1, 2, 3, 4, 12	syl13anc 1375	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑆 ∨ 𝐹) = (𝑆 ∨ 𝑈))
14		simp1l 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝐾 ∈ HL)
15		simp21 1208	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
16		simp23 1210	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑃 ≠ 𝑄)
17	5, 6, 7, 8, 9, 10	lhpat2 40505	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
18	1, 15, 3, 16, 17	syl112anc 1377	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑈 ∈ 𝐴)
19		simp31l 1298	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑆 ∈ 𝐴)
20	6, 8	hlatjcom 39828	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑈 ∨ 𝑆) = (𝑆 ∨ 𝑈))
21	14, 18, 19, 20	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑈 ∨ 𝑆) = (𝑆 ∨ 𝑈))
22	13, 21	eqtr4d 2775	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑆 ∨ 𝐹) = (𝑈 ∨ 𝑆))
23		simp32 1212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
24		cdleme12.g	. . . . . 6 ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
25	5, 6, 7, 8, 9, 10, 24	cdleme1 40687	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))) → (𝑇 ∨ 𝐺) = (𝑇 ∨ 𝑈))
26	1, 2, 3, 23, 25	syl13anc 1375	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑇 ∨ 𝐺) = (𝑇 ∨ 𝑈))
27		simp32l 1300	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑇 ∈ 𝐴)
28	6, 8	hlatjcom 39828	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑈 ∨ 𝑇) = (𝑇 ∨ 𝑈))
29	14, 18, 27, 28	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑈 ∨ 𝑇) = (𝑇 ∨ 𝑈))
30	26, 29	eqtr4d 2775	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑇 ∨ 𝐺) = (𝑈 ∨ 𝑇))
31	22, 30	oveq12d 7378	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝑆 ∨ 𝐹) ∧ (𝑇 ∨ 𝐺)) = ((𝑈 ∨ 𝑆) ∧ (𝑈 ∨ 𝑇)))
32		simp33 1213	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))
33	5, 6, 7, 8	2llnma2 40249	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑈 ∨ 𝑆) ∧ (𝑈 ∨ 𝑇)) = 𝑈)
34	14, 19, 27, 18, 32, 33	syl131anc 1386	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝑈 ∨ 𝑆) ∧ (𝑈 ∨ 𝑇)) = 𝑈)
35	31, 34	eqtrd 2772	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝑆 ∨ 𝐹) ∧ (𝑇 ∨ 𝐺)) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  lecple 17218  joincjn 18268  meetcmee 18269  Atomscatm 39723  HLchlt 39810  LHypclh 40444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39636  df-ol 39638  df-oml 39639  df-covers 39726  df-ats 39727  df-atl 39758  df-cvlat 39782  df-hlat 39811  df-psubsp 39963  df-pmap 39964  df-padd 40256  df-lhyp 40448

This theorem is referenced by:  cdleme13  40732  cdleme16b  40739