Theorem dihord4 42455

Description: The isomorphism H for a lattice 𝐾 is order-preserving in the region not under co-atom 𝑊. TODO: reformat (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) to eliminate adant*. (Contributed by NM, 6-Mar-2014.)

Hypotheses

Ref	Expression
dihord3.b	⊢ 𝐵 = (Base‘𝐾)
dihord3.l	⊢ ≤ = (le‘𝐾)
dihord3.h	⊢ 𝐻 = (LHyp‘𝐾)
dihord3.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

Assertion

Ref	Expression
dihord4	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))

Proof of Theorem dihord4

Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihord3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		dihord3.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		eqid 2763	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2763	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2763	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		dihord3.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7	1, 2, 3, 4, 5, 6	lhpmcvr2 41221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ (Atoms‘𝐾)(¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
8	7	3adant3 1150	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑞 ∈ (Atoms‘𝐾)(¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
9	1, 2, 3, 4, 5, 6	lhpmcvr2 41221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
10	9	3adant2 1149	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
11		reeanv 3237	. . 3 ⊢ (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) ↔ (∃𝑞 ∈ (Atoms‘𝐾)(¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)))
12	8, 10, 11	sylanbrc 595	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)))
13		simp11 1222	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		simp12 1223	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
15		simp2l 1218	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → 𝑞 ∈ (Atoms‘𝐾))
16		simp3ll 1263	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → ¬ 𝑞 ≤ 𝑊)
17	15, 16	jca 521	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 ≤ 𝑊))
18		simp3lr 1264	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)
19		dihord3.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
20		eqid 2763	. . . . . . . 8 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
21		eqid 2763	. . . . . . . 8 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
22		eqid 2763	. . . . . . . 8 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
23		eqid 2763	. . . . . . . 8 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
24	1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23	dihvalcq 42433	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐼‘𝑋) = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))))
25	13, 14, 17, 18, 24	syl112anc 1401	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝐼‘𝑋) = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))))
26		simp13 1224	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
27		simp2r 1219	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → 𝑟 ∈ (Atoms‘𝐾))
28		simp3rl 1265	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → ¬ 𝑟 ≤ 𝑊)
29	27, 28	jca 521	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊))
30		simp3rr 1266	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)
31	1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23	dihvalcq 42433	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
32	13, 26, 29, 30, 31	syl112anc 1401	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
33	25, 32	sseq12d 3945	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))))
34		simpl11 1267	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35		simpl2l 1245	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → 𝑞 ∈ (Atoms‘𝐾))
36	16	adantr 486	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → ¬ 𝑞 ≤ 𝑊)
37	35, 36	jca 521	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 ≤ 𝑊))
38		simpl2r 1246	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → 𝑟 ∈ (Atoms‘𝐾))
39	28	adantr 486	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → ¬ 𝑟 ≤ 𝑊)
40	38, 39	jca 521	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊))
41		simp12l 1305	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → 𝑋 ∈ 𝐵)
42	41	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → 𝑋 ∈ 𝐵)
43		simp13l 1307	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → 𝑌 ∈ 𝐵)
44	43	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → 𝑌 ∈ 𝐵)
45	18	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)
46	30	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)
47		simpr 490	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
48	1, 2, 3, 4, 5, 6, 20, 21, 22, 23	dihord2 42424	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))) → 𝑋 ≤ 𝑌)
49	34, 37, 40, 42, 44, 45, 46, 47, 48	syl323anc 1427	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) → 𝑋 ≤ 𝑌)
50		simpl11 1267	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51		simpl2l 1245	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → 𝑞 ∈ (Atoms‘𝐾))
52	16	adantr 486	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → ¬ 𝑞 ≤ 𝑊)
53	51, 52	jca 521	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 ≤ 𝑊))
54		simpl2r 1246	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → 𝑟 ∈ (Atoms‘𝐾))
55	28	adantr 486	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → ¬ 𝑟 ≤ 𝑊)
56	54, 55	jca 521	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊))
57	41	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
58	43	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
59	18	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)
60	30	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)
61		simpr 490	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
62	1, 2, 3, 4, 5, 6, 20, 21, 22, 23	dihord1 42415	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 ∧ 𝑋 ≤ 𝑌)) → ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
63	50, 53, 56, 57, 58, 59, 60, 61, 62	syl323anc 1427	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) ∧ 𝑋 ≤ 𝑌) → ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
64	49, 63	impbida 813	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → (((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))) ↔ 𝑋 ≤ 𝑌))
65	33, 64	bitrd 282	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))
66	65	3exp 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))))
67	66	rexlimdvv 3221	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)))
68	12, 67	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 1569   ∈ wcel 2146  ∃wrex 3089   ⊆ wss 3899   class class class wbr 5105  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  lecple 17447  joincjn 18497  meetcmee 18498  LSSumclsm 19860  Atomscatm 40460  HLchlt 40547  LHypclh 41181  DVecHcdvh 42275  DIsoBcdib 42335  DIsoCcdic 42369  DIsoHcdih 42425

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287  ax-riotaBAD 40150

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-iin 4956  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-tpos 8254  df-undef 8301  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-er 8728  df-map 8860  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-3 12416  df-4 12417  df-5 12418  df-6 12419  df-n0 12617  df-z 12704  df-uz 12976  df-fz 13650  df-struct 17337  df-sets 17354  df-slot 17372  df-ndx 17384  df-base 17400  df-ress 17421  df-plusg 17453  df-mulr 17454  df-sca 17456  df-vsca 17457  df-0g 17624  df-proset 18480  df-poset 18499  df-plt 18514  df-lub 18530  df-glb 18531  df-join 18532  df-meet 18533  df-p0 18609  df-p1 18610  df-lat 18618  df-clat 18685  df-mgm 18828  df-sgrp 18920  df-mnd 18936  df-submnd 18991  df-grp 19159  df-minusg 19160  df-sbg 19161  df-subg 19345  df-cntz 19543  df-lsm 19862  df-cmn 20008  df-abl 20009  df-mgp 20373  df-rng 20387  df-ur 20420  df-ring 20473  df-oppr 20579  df-dvdsr 20599  df-unit 20600  df-invr 20630  df-dvr 20643  df-drng 20994  df-lmod 21149  df-lss 21219  df-lsp 21259  df-lvec 21390  df-oposet 40373  df-ol 40375  df-oml 40376  df-covers 40463  df-ats 40464  df-atl 40495  df-cvlat 40519  df-hlat 40548  df-llines 40695  df-lplanes 40696  df-lvols 40697  df-lines 40698  df-psubsp 40700  df-pmap 40701  df-padd 40993  df-lhyp 41185  df-laut 41186  df-ldil 41301  df-ltrn 41302  df-trl 41356  df-tendo 41952  df-edring 41954  df-disoa 42226  df-dvech 42276  df-dib 42336  df-dic 42370  df-dih 42426

This theorem is used by:  dihord5apre  42459  dihord  42461