Theorem dihord5apre 39523

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 1190	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1192	. . 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 38285	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑌 ∧ 𝑊)) = 𝑌))
10	1, 2, 9	syl2anc 584	. 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 1283	. . . . . . . 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 37624	. . . . . . 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 1285	. . . . . . 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 1243	. . . . . . . . 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 37549	. . . . . . . . 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 18246	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑟 ∨ 𝑋) ∈ 𝐵)
18	12, 16, 13, 17	syl3anc 1370	. . . . . . 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 1287	. . . . . . 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 18256	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ (𝑟 ∨ 𝑋))
21	12, 16, 13, 20	syl3anc 1370	. . . . . . 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 1202	. . . . . . . . . 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 1244	. . . . . . . . . . 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 18255	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑟 ≤ (𝑟 ∨ 𝑋))
25	12, 16, 13, 24	syl3anc 1370	. . . . . . . . . . . 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 1284	. . . . . . . . . . . . . 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 38259	. . . . . . . . . . . . . 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 18251	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 692	. . . . . . . . . . 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 197	. . . . . . . . . 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 1200	. . . . . . . . . 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 1203	. . . . . . . . . . . 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 38303	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑟 ∨ 𝑋) ∧ 𝑊) = 𝑋)
36	22, 33, 34, 35	syl3anc 1370	. . . . . . . . . . 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 7345	. . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
40		eqid 2736	. . . . . . . . . . 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 39497	. . . . . . . . . 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 1378	. . . . . . . . 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 39371	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
47	46	lsssssubg 20318	. . . . . . . . . . . . . 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 39454	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘𝑈))
50	22, 33, 49	syl2anc 584	. . . . . . . . . . . . 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 3932	. . . . . . . . . . . 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 18247	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
53	12, 19, 28, 52	syl3anc 1370	. . . . . . . . . . . . . 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 18272	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ≤ 𝑊)
55	12, 19, 28, 54	syl3anc 1370	. . . . . . . . . . . . . 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 39431	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌 ∧ 𝑊)) ∈ (LSubSp‘𝑈))
57	22, 53, 55, 56	syl12anc 834	. . . . . . . . . . . . 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 3932	. . . . . . . . . . . 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 19350	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 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)) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
62		simp3r 1201	. . . . . . . . . . . 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 39497	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 3972	. . . . . . . . . 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 6823	. . . . . . . . . . . 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 39498	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
68	22, 34, 67	syl2anc 584	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 1136	. . . . . . . . . . 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 3969	. . . . . . . . . 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 18247	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ∈ 𝐵)
73	12, 18, 28, 72	syl3anc 1370	. . . . . . . . . . . . 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 18272	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑟 ∨ 𝑋) ∧ 𝑊) ≤ 𝑊)
75	12, 18, 28, 74	syl3anc 1370	. . . . . . . . . . . . 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 39431	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 3932	. . . . . . . . . . 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 39511	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
80	22, 19, 79	syl2anc 584	. . . . . . . . . . . 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 3932	. . . . . . . . . . 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 19357	. . . . . . . . . . 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 1370	. . . . . . . . . 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 709	. . . . . . . . 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 3969	. . . . . . . 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 39519	. . . . . . . . 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 1374	. . . . . . . 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 231	. . . . . . 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 18253	. . . . . 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 1120	. . . . 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 433	. . . 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 423	. . 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 3146	. 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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃wrex 3070   ⊆ wss 3897   class class class wbr 5089  ‘cfv 6473  (class class class)co 7329  Basecbs 17001  lecple 17058  joincjn 18118  meetcmee 18119  Latclat 18238  SubGrpcsubg 18837  LSSumclsm 19327  LModclmod 20221  LSubSpclss 20291  Atomscatm 37523  HLchlt 37610  LHypclh 38245  DVecHcdvh 39339  DIsoBcdib 39399  DIsoCcdic 39433  DIsoHcdih 39489

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041  ax-riotaBAD 37213

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-tpos 8104  df-undef 8151  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-struct 16937  df-sets 16954  df-slot 16972  df-ndx 16984  df-base 17002  df-ress 17031  df-plusg 17064  df-mulr 17065  df-sca 17067  df-vsca 17068  df-0g 17241  df-proset 18102  df-poset 18120  df-plt 18137  df-lub 18153  df-glb 18154  df-join 18155  df-meet 18156  df-p0 18232  df-p1 18233  df-lat 18239  df-clat 18306  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-submnd 18520  df-grp 18668  df-minusg 18669  df-sbg 18670  df-subg 18840  df-cntz 19011  df-lsm 19329  df-cmn 19475  df-abl 19476  df-mgp 19808  df-ur 19825  df-ring 19872  df-oppr 19949  df-dvdsr 19970  df-unit 19971  df-invr 20001  df-dvr 20012  df-drng 20087  df-lmod 20223  df-lss 20292  df-lsp 20332  df-lvec 20463  df-oposet 37436  df-ol 37438  df-oml 37439  df-covers 37526  df-ats 37527  df-atl 37558  df-cvlat 37582  df-hlat 37611  df-llines 37759  df-lplanes 37760  df-lvols 37761  df-lines 37762  df-psubsp 37764  df-pmap 37765  df-padd 38057  df-lhyp 38249  df-laut 38250  df-ldil 38365  df-ltrn 38366  df-trl 38420  df-tendo 39016  df-edring 39018  df-disoa 39290  df-dvech 39340  df-dib 39400  df-dic 39434  df-dih 39490

This theorem is referenced by:  dihord5a  39524