Theorem dihord5apre 41719

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 1193	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1195	. . 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 40481	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌))
10	1, 2, 9	syl2anc 585	. 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 1286	. . . . . . . 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 39821	. . . . . . 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 1288	. . . . . . 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 1246	. . . . . . . . 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 39746	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵)
16	14, 15	syl 17	. . . . . . . 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 18394	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑟 ∨ 𝑋) ∈ 𝐵)
18	12, 16, 13, 17	syl3anc 1374	. . . . . . 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 1290	. . . . . . 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 18404	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ (𝑟 ∨ 𝑋))
21	12, 16, 13, 20	syl3anc 1374	. . . . . . 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 1205	. . . . . . . . . 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 1247	. . . . . . . . . . 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 18403	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑟 ≤ (𝑟 ∨ 𝑋))
25	12, 16, 13, 24	syl3anc 1374	. . . . . . . . . . . 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 1287	. . . . . . . . . . . . . 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 40455	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
28	26, 27	syl 17	. . . . . . . . . . . . 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 18399	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 696	. . . . . . . . . . 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 198	. . . . . . . . . 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 1203	. . . . . . . . . 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 1206	. . . . . . . . . . . 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 40499	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑟 ∨ 𝑋) ∧ 𝑊) = 𝑋)
36	22, 33, 34, 35	syl3anc 1374	. . . . . . . . . . 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 7374	. . . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
40		eqid 2737	. . . . . . . . . . 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 41693	. . . . . . . . . 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 1382	. . . . . . . . 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 41567	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
47	46	lsssssubg 20942	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
48	45, 47	syl 17	. . . . . . . . . . . . 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 41650	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘𝑈))
50	22, 33, 49	syl2anc 585	. . . . . . . . . . . . 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 3923	. . . . . . . . . . . 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 18395	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
53	12, 19, 28, 52	syl3anc 1374	. . . . . . . . . . . . . 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 18420	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ≤ 𝑊)
55	12, 19, 28, 54	syl3anc 1374	. . . . . . . . . . . . . 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 41627	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
57	22, 53, 55, 56	syl12anc 837	. . . . . . . . . . . . 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 3923	. . . . . . . . . . . 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 19621	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 1207	. . . . . . . . . . . 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 1204	. . . . . . . . . . . 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 41693	. . . . . . . . . . . 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 1377	. . . . . . . . . . 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 3960	. . . . . . . . . 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 6836	. . . . . . . . . . . 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 41694	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
68	22, 34, 67	syl2anc 585	. . . . . . . . . . . 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 2775	. . . . . . . . . . 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 1138	. . . . . . . . . . 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 3957	. . . . . . . . . 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 18395	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ∈ 𝐵)
73	12, 18, 28, 72	syl3anc 1374	. . . . . . . . . . . . 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 18420	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ≤ 𝑊)
75	12, 18, 28, 74	syl3anc 1374	. . . . . . . . . . . . 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 41627	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 3923	. . . . . . . . . . 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 41707	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
80	22, 19, 79	syl2anc 585	. . . . . . . . . . . 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 3923	. . . . . . . . . . 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 19628	. . . . . . . . . . 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 1374	. . . . . . . . . 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 713	. . . . . . . . 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 3957	. . . . . . . 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 41715	. . . . . . . . 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 1378	. . . . . . . 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 232	. . . . . . 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 18401	. . . . . 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 1122	. . . . 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 432	. . . 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 422	. . 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 3137	. 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 15	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890   class class class wbr 5086  ‘cfv 6490  (class class class)co 7358  Basecbs 17168  lecple 17216  joincjn 18266  meetcmee 18267  Latclat 18386  SubGrpcsubg 19085  LSSumclsm 19598  LModclmod 20844  LSubSpclss 20915  Atomscatm 39720  HLchlt 39807  LHypclh 40441  DVecHcdvh 41535  DIsoBcdib 41595  DIsoCcdic 41629  DIsoHcdih 41685

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 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-riotaBAD 39410

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  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-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8167  df-undef 8214  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-n0 12427  df-z 12514  df-uz 12778  df-fz 13451  df-struct 17106  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-mulr 17223  df-sca 17225  df-vsca 17226  df-0g 17393  df-proset 18249  df-poset 18268  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18387  df-clat 18454  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-submnd 18741  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19088  df-cntz 19281  df-lsm 19600  df-cmn 19746  df-abl 19747  df-mgp 20111  df-rng 20123  df-ur 20152  df-ring 20205  df-oppr 20306  df-dvdsr 20326  df-unit 20327  df-invr 20357  df-dvr 20370  df-drng 20697  df-lmod 20846  df-lss 20916  df-lsp 20956  df-lvec 21088  df-oposet 39633  df-ol 39635  df-oml 39636  df-covers 39723  df-ats 39724  df-atl 39755  df-cvlat 39779  df-hlat 39808  df-llines 39955  df-lplanes 39956  df-lvols 39957  df-lines 39958  df-psubsp 39960  df-pmap 39961  df-padd 40253  df-lhyp 40445  df-laut 40446  df-ldil 40561  df-ltrn 40562  df-trl 40616  df-tendo 41212  df-edring 41214  df-disoa 41486  df-dvech 41536  df-dib 41596  df-dic 41630  df-dih 41686

This theorem is referenced by:  dihord5a  41720