Theorem cdlemn2 41240

Description: Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)

Hypotheses

Ref	Expression
cdlemn2.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn2.l	⊢ ≤ = (le‘𝐾)
cdlemn2.j	⊢ ∨ = (join‘𝐾)
cdlemn2.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn2.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn2.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemn2.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑆)

Assertion

Ref	Expression
cdlemn2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ 𝑋)

Distinct variable groups:   ≤ ,ℎ   𝐴,ℎ   ℎ,𝐻   ℎ,𝐾   𝑄,ℎ   𝑆,ℎ   𝑇,ℎ   ℎ,𝑊

Allowed substitution hints:   𝐵(ℎ)   𝑅(ℎ)   𝐹(ℎ)   ∨ (ℎ)   𝑋(ℎ)

Proof of Theorem cdlemn2

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp21 1207	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
3		simp22 1208	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
4		cdlemn2.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
5		cdlemn2.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemn2.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemn2.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemn2.f	. . . . . . 7 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑆)
9	4, 5, 6, 7, 8	ltrniotacl 40624	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
10	1, 2, 3, 9	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐹 ∈ 𝑇)
11		cdlemn2.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
12		eqid 2731	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
13		cdlemn2.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14	4, 11, 12, 5, 6, 7, 13	trlval2 40208	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑄 ∨ (𝐹‘𝑄))(meet‘𝐾)𝑊))
15	1, 10, 2, 14	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) = ((𝑄 ∨ (𝐹‘𝑄))(meet‘𝐾)𝑊))
16	4, 5, 6, 7, 8	ltrniotaval 40626	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝐹‘𝑄) = 𝑆)
17	1, 2, 3, 16	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐹‘𝑄) = 𝑆)
18	17	oveq2d 7362	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ (𝐹‘𝑄)) = (𝑄 ∨ 𝑆))
19	18	oveq1d 7361	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ (𝐹‘𝑄))(meet‘𝐾)𝑊) = ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊))
20	15, 19	eqtrd 2766	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) = ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊))
21		simp1l 1198	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐾 ∈ HL)
22	21	hllatd 39409	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐾 ∈ Lat)
23		simp21l 1291	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑄 ∈ 𝐴)
24		cdlemn2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
25	24, 5	atbase 39334	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
26	23, 25	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑄 ∈ 𝐵)
27		simp23l 1295	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑋 ∈ 𝐵)
28	24, 4, 11	latlej1 18354	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑄 ≤ (𝑄 ∨ 𝑋))
29	22, 26, 27, 28	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑄 ≤ (𝑄 ∨ 𝑋))
30		simp3 1138	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑆 ≤ (𝑄 ∨ 𝑋))
31		simp22l 1293	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑆 ∈ 𝐴)
32	24, 5	atbase 39334	. . . . . . 7 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵)
33	31, 32	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑆 ∈ 𝐵)
34	24, 11	latjcl 18345	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) ∈ 𝐵)
35	22, 26, 27, 34	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ 𝑋) ∈ 𝐵)
36	24, 4, 11	latjle12 18356	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵 ∧ (𝑄 ∨ 𝑋) ∈ 𝐵)) → ((𝑄 ≤ (𝑄 ∨ 𝑋) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) ↔ (𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋)))
37	22, 26, 33, 35, 36	syl13anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ≤ (𝑄 ∨ 𝑋) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) ↔ (𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋)))
38	29, 30, 37	mpbi2and 712	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋))
39	24, 11, 5	hlatjcl 39412	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑄 ∨ 𝑆) ∈ 𝐵)
40	21, 23, 31, 39	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ 𝑆) ∈ 𝐵)
41		simp1r 1199	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑊 ∈ 𝐻)
42	24, 6	lhpbase 40043	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
43	41, 42	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑊 ∈ 𝐵)
44	24, 4, 12	latmlem1 18375	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ((𝑄 ∨ 𝑆) ∈ 𝐵 ∧ (𝑄 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊)))
45	22, 40, 35, 43, 44	syl13anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊)))
46	38, 45	mpd 15	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊))
47	20, 46	eqbrtrd 5113	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊))
48		simp23 1209	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
49	24, 4, 11, 12, 5, 6	lhple 40087	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊) = 𝑋)
50	1, 2, 48, 49	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊) = 𝑋)
51	47, 50	breqtrd 5117	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   class class class wbr 5091  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  Latclat 18337  Atomscatm 39308  HLchlt 39395  LHypclh 40029  LTrncltrn 40146  trLctrl 40203

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-riotaBAD 38998

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-undef 8203  df-map 8752  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39221  df-ol 39223  df-oml 39224  df-covers 39311  df-ats 39312  df-atl 39343  df-cvlat 39367  df-hlat 39396  df-llines 39543  df-lplanes 39544  df-lvols 39545  df-lines 39546  df-psubsp 39548  df-pmap 39549  df-padd 39841  df-lhyp 40033  df-laut 40034  df-ldil 40149  df-ltrn 40150  df-trl 40204

This theorem is referenced by:  cdlemn2a  41241