Theorem dihord5apre 42143

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Hypotheses

Ref	Expression
dihord5apre.b	⊢ 𝐵 = (Base‘𝐾)
dihord5apre.l	⊢ ≤ = (le‘𝐾)
dihord5apre.h	⊢ 𝐻 = (LHyp‘𝐾)
dihord5apre.j	⊢ ∨ = (join‘𝐾)
dihord5apre.m	⊢ ∧ = (meet‘𝐾)
dihord5apre.a	⊢ 𝐴 = (Atoms‘𝐾)
dihord5apre.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihord5apre.s	⊢ ⊕ = (LSSum‘𝑈)
dihord5apre.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

Assertion

Ref	Expression
dihord5apre	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)

Proof of Theorem dihord5apre

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1210	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1212	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
3		dihord5apre.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
4		dihord5apre.l	. . . 4 ⊢ ≤ = (le‘𝐾)
5		dihord5apre.j	. . . 4 ⊢ ∨ = (join‘𝐾)
6		dihord5apre.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
7		dihord5apre.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
8		dihord5apre.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
9	3, 4, 5, 6, 7, 8	lhpmcvr2 40905	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌))
10	1, 2, 9	syl2anc 596	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌))
11		simp11l 1303	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝐾 ∈ HL)
12	11	hllatd 40245	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝐾 ∈ Lat)
13		simp12l 1305	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑋 ∈ 𝐵)
14		simp3ll 1263	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑟 ∈ 𝐴)
15	3, 7	atbase 40170	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵)
16	14, 15	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑟 ∈ 𝐵)
17	3, 5	latjcl 18533	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑟 ∨ 𝑋) ∈ 𝐵)
18	12, 16, 13, 17	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑟 ∨ 𝑋) ∈ 𝐵)
19		simp13l 1307	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑌 ∈ 𝐵)
20	3, 4, 5	latlej2 18543	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ (𝑟 ∨ 𝑋))
21	12, 16, 13, 20	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑋 ≤ (𝑟 ∨ 𝑋))
22		simp11 1222	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		simp3lr 1264	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ¬ 𝑟 ≤ 𝑊)
24	3, 4, 5	latlej1 18542	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑟 ≤ (𝑟 ∨ 𝑋))
25	12, 16, 13, 24	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑟 ≤ (𝑟 ∨ 𝑋))
26		simp11r 1304	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑊 ∈ 𝐻)
27	3, 8	lhpbase 40879	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
28	26, 27	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑊 ∈ 𝐵)
29	3, 4	lattr 18538	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ 𝐵 ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑟 ≤ (𝑟 ∨ 𝑋) ∧ (𝑟 ∨ 𝑋) ≤ 𝑊) → 𝑟 ≤ 𝑊))
30	12, 16, 18, 28, 29	syl13anc 1399	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((𝑟 ≤ (𝑟 ∨ 𝑋) ∧ (𝑟 ∨ 𝑋) ≤ 𝑊) → 𝑟 ≤ 𝑊))
31	25, 30	mpand 708	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((𝑟 ∨ 𝑋) ≤ 𝑊 → 𝑟 ≤ 𝑊))
32	23, 31	mtod 201	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ¬ (𝑟 ∨ 𝑋) ≤ 𝑊)
33		simp3l 1220	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
34		simp12 1223	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
35	3, 4, 5, 6, 7, 8	lhple 40923	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑟 ∨ 𝑋) ∧ 𝑊) = 𝑋)
36	22, 33, 34, 35	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((𝑟 ∨ 𝑋) ∧ 𝑊) = 𝑋)
37	36	oveq2d 7433	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑟 ∨ ((𝑟 ∨ 𝑋) ∧ 𝑊)) = (𝑟 ∨ 𝑋))
38		dihord5apre.i	. . . . . . . . . . 11 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
39		eqid 2762	. . . . . . . . . . 11 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
40		eqid 2762	. . . . . . . . . . 11 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
41		dihord5apre.u	. . . . . . . . . . 11 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
42		dihord5apre.s	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈)
43	3, 4, 5, 6, 7, 8, 38, 39, 40, 41, 42	dihvalcq 42117	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∨ 𝑋) ∈ 𝐵 ∧ ¬ (𝑟 ∨ 𝑋) ≤ 𝑊) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ ((𝑟 ∨ 𝑋) ∧ 𝑊)) = (𝑟 ∨ 𝑋))) → (𝐼‘(𝑟 ∨ 𝑋)) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊))))
44	22, 18, 32, 33, 37, 43	syl122anc 1406	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘(𝑟 ∨ 𝑋)) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊))))
45	8, 41, 22	dvhlmod 41991	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑈 ∈ LMod)
46		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
47	46	lsssssubg 21148	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
48	45, 47	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
49	4, 7, 8, 41, 40, 46	diclss 42074	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘𝑈))
50	22, 33, 49	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘𝑈))
51	48, 50	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘𝑈))
52	3, 6	latmcl 18534	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
53	12, 19, 28, 52	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑌 ∧ 𝑊) ∈ 𝐵)
54	3, 4, 6	latmle2 18559	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ≤ 𝑊)
55	12, 19, 28, 54	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑌 ∧ 𝑊) ≤ 𝑊)
56	3, 4, 8, 41, 39, 46	diblss 42051	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
57	22, 53, 55, 56	syl12anc 850	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
58	48, 57	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊)) ∈ (SubGrp‘𝑈))
59	42	lsmub1 19790	. . . . . . . . . . . 12 ⊢ (((((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘𝑈) ∧ (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊)) ∈ (SubGrp‘𝑈)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊))))
60	51, 58, 59	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊))))
61		simp13 1224	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
62		simp3r 1221	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)
63	3, 4, 5, 6, 7, 8, 38, 39, 40, 41, 42	dihvalcq 42117	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊))))
64	22, 61, 33, 62, 63	syl112anc 1401	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊))))
65	60, 64	sseqtrrd 3971	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊆ (𝐼‘𝑌))
66	36	fveq2d 6886	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
67	3, 4, 8, 38, 39	dihvalb 42118	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
68	22, 34, 67	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
69	66, 68	eqtr4d 2800	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) = (𝐼‘𝑋))
70		simp2 1155	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))
71	69, 70	eqsstrd 3968	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ⊆ (𝐼‘𝑌))
72	3, 6	latmcl 18534	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ∈ 𝐵)
73	12, 18, 28, 72	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ∈ 𝐵)
74	3, 4, 6	latmle2 18559	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ≤ 𝑊)
75	12, 18, 28, 74	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ≤ 𝑊)
76	3, 4, 8, 41, 39, 46	diblss 42051	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((𝑟 ∨ 𝑋) ∧ 𝑊) ∈ 𝐵 ∧ ((𝑟 ∨ 𝑋) ∧ 𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ∈ (LSubSp‘𝑈))
77	22, 73, 75, 76	syl12anc 850	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ∈ (LSubSp‘𝑈))
78	48, 77	sseldd 3935	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ∈ (SubGrp‘𝑈))
79	3, 8, 38, 41, 46	dihlss 42131	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
80	22, 19, 79	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
81	48, 80	sseldd 3935	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘𝑌) ∈ (SubGrp‘𝑈))
82	42	lsmlub 19797	. . . . . . . . . . 11 ⊢ (((((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘𝑈) ∧ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑌) ∈ (SubGrp‘𝑈)) → (((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊆ (𝐼‘𝑌) ∧ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ⊆ (𝐼‘𝑌)) ↔ ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊))) ⊆ (𝐼‘𝑌)))
83	51, 78, 81, 82	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊆ (𝐼‘𝑌) ∧ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊)) ⊆ (𝐼‘𝑌)) ↔ ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊))) ⊆ (𝐼‘𝑌)))
84	65, 71, 83	mpbi2and 725	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((((DIsoC‘𝐾)‘𝑊)‘𝑟) ⊕ (((DIsoB‘𝐾)‘𝑊)‘((𝑟 ∨ 𝑋) ∧ 𝑊))) ⊆ (𝐼‘𝑌))
85	44, 84	eqsstrd 3968	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝐼‘(𝑟 ∨ 𝑋)) ⊆ (𝐼‘𝑌))
86	3, 4, 8, 38	dihord4 42139	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∨ 𝑋) ∈ 𝐵 ∧ ¬ (𝑟 ∨ 𝑋) ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘(𝑟 ∨ 𝑋)) ⊆ (𝐼‘𝑌) ↔ (𝑟 ∨ 𝑋) ≤ 𝑌))
87	22, 18, 32, 61, 86	syl121anc 1402	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → ((𝐼‘(𝑟 ∨ 𝑋)) ⊆ (𝐼‘𝑌) ↔ (𝑟 ∨ 𝑋) ≤ 𝑌))
88	85, 87	mpbid 235	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → (𝑟 ∨ 𝑋) ≤ 𝑌)
89	3, 4, 12, 13, 18, 19, 21, 88	lattrd 18540	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑋 ≤ 𝑌)
90	89	3expia 1139	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌) → 𝑋 ≤ 𝑌))
91	90	exp4c 438	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (𝑟 ∈ 𝐴 → (¬ 𝑟 ≤ 𝑊 → ((𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌 → 𝑋 ≤ 𝑌))))
92	91	imp4a 428	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (𝑟 ∈ 𝐴 → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌) → 𝑋 ≤ 𝑌)))
93	92	rexlimdv 3163	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌) → 𝑋 ≤ 𝑌))
94	10, 93	mpd 16	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  lecple 17355  joincjn 18405  meetcmee 18406  Latclat 18525  SubGrpcsubg 19249  LSSumclsm 19767  LModclmod 21050  LSubSpclss 21121  Atomscatm 40144  HLchlt 40231  LHypclh 40865  DVecHcdvh 41959  DIsoBcdib 42019  DIsoCcdic 42053  DIsoHcdih 42109

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-riotaBAD 39834

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-tpos 8228  df-undef 8275  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-0g 17532  df-proset 18388  df-poset 18407  df-plt 18422  df-lub 18438  df-glb 18439  df-join 18440  df-meet 18441  df-p0 18517  df-p1 18518  df-lat 18526  df-clat 18593  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-grp 19066  df-minusg 19067  df-sbg 19068  df-subg 19252  df-cntz 19450  df-lsm 19769  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-invr 20535  df-dvr 20548  df-drng 20898  df-lmod 21052  df-lss 21122  df-lsp 21162  df-lvec 21293  df-oposet 40057  df-ol 40059  df-oml 40060  df-covers 40147  df-ats 40148  df-atl 40179  df-cvlat 40203  df-hlat 40232  df-llines 40379  df-lplanes 40380  df-lvols 40381  df-lines 40382  df-psubsp 40384  df-pmap 40385  df-padd 40677  df-lhyp 40869  df-laut 40870  df-ldil 40985  df-ltrn 40986  df-trl 41040  df-tendo 41636  df-edring 41638  df-disoa 41910  df-dvech 41960  df-dib 42020  df-dic 42054  df-dih 42110

This theorem is used by:  dihord5a  42144