Theorem dihord1 40545

Description: Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 to 𝑄 ≤ 𝑋 using lhpmcvr3 39352, here and all theorems below. (Contributed by NM, 2-Mar-2014.)

Hypotheses

Ref	Expression
dihjust.b	⊢ 𝐵 = (Base‘𝐾)
dihjust.l	⊢ ≤ = (le‘𝐾)
dihjust.j	⊢ ∨ = (join‘𝐾)
dihjust.m	⊢ ∧ = (meet‘𝐾)
dihjust.a	⊢ 𝐴 = (Atoms‘𝐾)
dihjust.h	⊢ 𝐻 = (LHyp‘𝐾)
dihjust.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
dihjust.J	⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
dihjust.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjust.s	⊢ ⊕ = (LSSum‘𝑈)

Assertion

Ref	Expression
dihord1	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))

Proof of Theorem dihord1

Step	Hyp	Ref	Expression
1		simp11 1200	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp13 1202	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
3		simp12 1201	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simp11l 1281	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL)
5	4	hllatd 38690	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Lat)
6		simp2r 1197	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝐵)
7		simp11r 1282	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑊 ∈ 𝐻)
8		dihjust.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
9		dihjust.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
10	8, 9	lhpbase 39325	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
11	7, 10	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑊 ∈ 𝐵)
12		dihjust.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
13	8, 12	latmcl 18392	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
14	5, 6, 11, 13	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ 𝑊) ∈ 𝐵)
15		dihjust.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
16	8, 15, 12	latmle2 18417	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ≤ 𝑊)
17	5, 6, 11, 16	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ 𝑊) ≤ 𝑊)
18	14, 17	jca 511	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊))
19		simp12l 1283	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑄 ∈ 𝐴)
20		dihjust.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
21	8, 20	atbase 38615	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
22	19, 21	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑄 ∈ 𝐵)
23		simp2l 1196	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
24	8, 12	latmcl 18392	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
25	5, 23, 11, 24	syl3anc 1368	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
26		dihjust.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
27	8, 26	latjcl 18391	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → (𝑄 ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵)
28	5, 22, 25, 27	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑄 ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵)
29	8, 15, 26	latlej1 18400	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑄 ≤ (𝑄 ∨ (𝑋 ∧ 𝑊)))
30	5, 22, 25, 29	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑄 ≤ (𝑄 ∨ (𝑋 ∧ 𝑊)))
31		simp31 1206	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
32		simp33 1208	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
33	31, 32	eqbrtrd 5160	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑄 ∨ (𝑋 ∧ 𝑊)) ≤ 𝑌)
34	8, 15, 5, 22, 28, 6, 30, 33	lattrd 18398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑄 ≤ 𝑌)
35		simp32 1207	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌)
36	34, 35	breqtrrd 5166	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑄 ≤ (𝑅 ∨ (𝑌 ∧ 𝑊)))
37		dihjust.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
38		dihjust.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
39		dihjust.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
40		dihjust.J	. . . 4 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
41	8, 15, 26, 20, 9, 37, 38, 39, 40	cdlemn5 40528	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) ∧ 𝑄 ≤ (𝑅 ∨ (𝑌 ∧ 𝑊))) → (𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))
42	1, 2, 3, 18, 36, 41	syl131anc 1380	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))
43	8, 15, 12	latmlem1 18421	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
44	5, 23, 6, 11, 43	syl13anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
45	32, 44	mpd 15	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊))
46	8, 15, 12	latmle2 18417	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
47	5, 23, 11, 46	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ∧ 𝑊) ≤ 𝑊)
48	8, 15, 9, 39	dibord 40486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → ((𝐼‘(𝑋 ∧ 𝑊)) ⊆ (𝐼‘(𝑌 ∧ 𝑊)) ↔ (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
49	1, 25, 47, 14, 17, 48	syl122anc 1376	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((𝐼‘(𝑋 ∧ 𝑊)) ⊆ (𝐼‘(𝑌 ∧ 𝑊)) ↔ (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
50	45, 49	mpbird 257	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑊)) ⊆ (𝐼‘(𝑌 ∧ 𝑊)))
51	9, 37, 1	dvhlmod 40437	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → 𝑈 ∈ LMod)
52		eqid 2724	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
53	52	lsssssubg 20790	. . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
54	51, 53	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
55	15, 20, 9, 37, 40, 52	diclss 40520	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐽‘𝑅) ∈ (LSubSp‘𝑈))
56	1, 2, 55	syl2anc 583	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐽‘𝑅) ∈ (LSubSp‘𝑈))
57	54, 56	sseldd 3975	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐽‘𝑅) ∈ (SubGrp‘𝑈))
58	8, 15, 9, 37, 39, 52	diblss 40497	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → (𝐼‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
59	1, 14, 17, 58	syl12anc 834	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
60	54, 59	sseldd 3975	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑌 ∧ 𝑊)) ∈ (SubGrp‘𝑈))
61	38	lsmub2 19563	. . . 4 ⊢ (((𝐽‘𝑅) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑌 ∧ 𝑊)) ∈ (SubGrp‘𝑈)) → (𝐼‘(𝑌 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))
62	57, 60, 61	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑌 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))
63	50, 62	sstrd 3984	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))
64	15, 20, 9, 37, 40, 52	diclss 40520	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐽‘𝑄) ∈ (LSubSp‘𝑈))
65	1, 3, 64	syl2anc 583	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐽‘𝑄) ∈ (LSubSp‘𝑈))
66	54, 65	sseldd 3975	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐽‘𝑄) ∈ (SubGrp‘𝑈))
67	8, 15, 9, 37, 39, 52	diblss 40497	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
68	1, 25, 47, 67	syl12anc 834	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
69	54, 68	sseldd 3975	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑊)) ∈ (SubGrp‘𝑈))
70	52, 38	lsmcl 20916	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑅) ∈ (LSubSp‘𝑈) ∧ (𝐼‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈)) → ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∈ (LSubSp‘𝑈))
71	51, 56, 59, 70	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∈ (LSubSp‘𝑈))
72	54, 71	sseldd 3975	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∈ (SubGrp‘𝑈))
73	38	lsmlub 19569	. . 3 ⊢ (((𝐽‘𝑄) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ∈ (SubGrp‘𝑈) ∧ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∈ (SubGrp‘𝑈)) → (((𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ↔ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))))
74	66, 69, 72, 73	syl3anc 1368	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → (((𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ↔ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))))
75	42, 63, 74	mpbi2and 709	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3940   class class class wbr 5138  ‘cfv 6533  (class class class)co 7401  Basecbs 17140  lecple 17200  joincjn 18263  meetcmee 18264  Latclat 18383  SubGrpcsubg 19032  LSSumclsm 19539  LModclmod 20691  LSubSpclss 20763  Atomscatm 38589  HLchlt 38676  LHypclh 39311  DVecHcdvh 40405  DIsoBcdib 40465  DIsoCcdic 40499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-riotaBAD 38279

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-undef 8253  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-0g 17383  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-mgm 18560  df-sgrp 18639  df-mnd 18655  df-submnd 18701  df-grp 18853  df-minusg 18854  df-sbg 18855  df-subg 19035  df-cntz 19218  df-lsm 19541  df-cmn 19687  df-abl 19688  df-mgp 20025  df-rng 20043  df-ur 20072  df-ring 20125  df-oppr 20221  df-dvdsr 20244  df-unit 20245  df-invr 20275  df-dvr 20288  df-drng 20574  df-lmod 20693  df-lss 20764  df-lsp 20804  df-lvec 20936  df-oposet 38502  df-ol 38504  df-oml 38505  df-covers 38592  df-ats 38593  df-atl 38624  df-cvlat 38648  df-hlat 38677  df-llines 38825  df-lplanes 38826  df-lvols 38827  df-lines 38828  df-psubsp 38830  df-pmap 38831  df-padd 39123  df-lhyp 39315  df-laut 39316  df-ldil 39431  df-ltrn 39432  df-trl 39486  df-tendo 40082  df-edring 40084  df-disoa 40356  df-dvech 40406  df-dib 40466  df-dic 40500

This theorem is referenced by:  dihord4  40585