Theorem dihord5apre 42077

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 40839	. . 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 40179	. . . . . . 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 40104	. . . . . . . . 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 18520	. . . . . . . 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 18530	. . . . . . . 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 18529	. . . . . . . . . . . . 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 40813	. . . . . . . . . . . . . 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 18525	. . . . . . . . . . . . 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 40857	. . . . . . . . . . . 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 7439	. . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
40		eqid 2766	. . . . . . . . . . 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 42051	. . . . . . . . . 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 41925	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 15 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
47	46	lsssssubg 21116	. . . . . . . . . . . . . 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 42008	. . . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 18521	. . . . . . . . . . . . . . 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 18546	. . . . . . . . . . . . . . 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 41985	. . . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 19758	. . . . . . . . . . . 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 42051	. . . . . . . . . . . 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 3977	. . . . . . . . . 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 6892	. . . . . . . . . . . 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 42052	. . . . . . . . . . . . 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 2804	. . . . . . . . . . 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 3974	. . . . . . . . . 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 18521	. . . . . . . . . . . . . 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 18546	. . . . . . . . . . . . . 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 41985	. . . . . . . . . . . . 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 3941	. . . . . . . . . . 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 42065	. . . . . . . . . . . . 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 3941	. . . . . . . . . . 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 19765	. . . . . . . . . . 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 3974	. . . . . . . 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 42073	. . . . . . . . 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 18527	. . . . . 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 3167	. 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 2146  ∃wrex 3092   ⊆ wss 3908   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  lecple 17342  joincjn 18392  meetcmee 18393  Latclat 18512  SubGrpcsubg 19217  LSSumclsm 19735  LModclmod 21018  LSubSpclss 21089  Atomscatm 40078  HLchlt 40165  LHypclh 40799  DVecHcdvh 41893  DIsoBcdib 41953  DIsoCcdic 41987  DIsoHcdih 42043

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-riotaBAD 39768

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  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 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-tpos 8231  df-undef 8278  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-0g 17519  df-proset 18375  df-poset 18394  df-plt 18409  df-lub 18425  df-glb 18426  df-join 18427  df-meet 18428  df-p0 18504  df-p1 18505  df-lat 18513  df-clat 18580  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-submnd 18873  df-grp 19034  df-minusg 19035  df-sbg 19036  df-subg 19220  df-cntz 19418  df-lsm 19737  df-cmn 19883  df-abl 19884  df-mgp 20248  df-rng 20262  df-ur 20295  df-ring 20348  df-oppr 20452  df-dvdsr 20472  df-unit 20473  df-invr 20503  df-dvr 20516  df-drng 20866  df-lmod 21020  df-lss 21090  df-lsp 21130  df-lvec 21261  df-oposet 39991  df-ol 39993  df-oml 39994  df-covers 40081  df-ats 40082  df-atl 40113  df-cvlat 40137  df-hlat 40166  df-llines 40313  df-lplanes 40314  df-lvols 40315  df-lines 40316  df-psubsp 40318  df-pmap 40319  df-padd 40611  df-lhyp 40803  df-laut 40804  df-ldil 40919  df-ltrn 40920  df-trl 40974  df-tendo 41570  df-edring 41572  df-disoa 41844  df-dvech 41894  df-dib 41954  df-dic 41988  df-dih 42044

This theorem is used by:  dihord5a  42078