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Theorem dihord4 37040
Description: The isomorphism H for a lattice 𝐾 is order-preserving in the region not under co-atom 𝑊. TODO: reformat q e. 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))

Proof of Theorem dihord4
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihord3.b . . . . 5 𝐵 = (Base‘𝐾)
2 dihord3.l . . . . 5 = (le‘𝐾)
3 eqid 2813 . . . . 5 (join‘𝐾) = (join‘𝐾)
4 eqid 2813 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
5 eqid 2813 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
6 dihord3.h . . . . 5 𝐻 = (LHyp‘𝐾)
71, 2, 3, 4, 5, 6lhpmcvr2 35806 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑞 ∈ (Atoms‘𝐾)(¬ 𝑞 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
873adant3 1155 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ∃𝑞 ∈ (Atoms‘𝐾)(¬ 𝑞 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
91, 2, 3, 4, 5, 6lhpmcvr2 35806 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
1093adant2 1154 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
11 reeanv 3302 . . 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)))
128, 10, 11sylanbrc 574 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1253 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 ∧ 𝑊𝐻))
14 simp12 1254 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
15 simp2l 1249 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
1715, 16jca 503 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1319 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 2813 . . . . . . . 8 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
21 eqid 2813 . . . . . . . 8 ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
22 eqid 2813 . . . . . . . 8 ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
23 eqid 2813 . . . . . . . 8 (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
241, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23dihvalcq 37018 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐼𝑋) = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))))
2513, 14, 17, 18, 24syl112anc 1486 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → (𝐼𝑋) = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))))
26 simp13 1255 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → (𝑌𝐵 ∧ ¬ 𝑌 𝑊))
27 simp2r 1250 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1320 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
2927, 28jca 503 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1321 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
311, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23dihvalcq 37018 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊) ∧ ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) → (𝐼𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
3213, 26, 29, 30, 31syl112anc 1486 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → (𝐼𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
3325, 32sseq12d 3838 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ ((((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 1322 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 ∧ 𝑊𝐻))
35 simpl2l 1290 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))
3616adantr 468 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
3735, 36jca 503 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1292 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))
3928adantr 468 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
4038, 39jca 503 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1378 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → 𝑋𝐵)
4241adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)))) → 𝑋𝐵)
43 simp13l 1380 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → 𝑌𝐵)
4443adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)))) → 𝑌𝐵)
4518adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
4630adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 473 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))))
481, 2, 3, 4, 5, 6, 20, 21, 22, 23dihord2 37009 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 𝑊) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((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)
4934, 37, 40, 42, 44, 45, 46, 47, 48syl323anc 1512 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1322 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 ∧ 𝑊𝐻))
51 simpl2l 1290 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))
5216adantr 468 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
5351, 52jca 503 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 1292 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))
5528adantr 468 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
5654, 55jca 503 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))
5741adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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) → 𝑋𝐵)
5843adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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) → 𝑌𝐵)
5918adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
6030adantr 468 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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 473 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)
621, 2, 3, 4, 5, 6, 20, 21, 22, 23dihord1 37000 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞 𝑊) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌𝑋 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))))
6350, 53, 56, 57, 58, 59, 60, 61, 62syl323anc 1512 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))))
6449, 63impbida 826 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))
6533, 64bitrd 270 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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))) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))
66653exp 1141 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))))
6766rexlimdvv 3232 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 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)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌)))
6812, 67mpd 15 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wrex 3104  wss 3776   class class class wbr 4851  cfv 6104  (class class class)co 6877  Basecbs 16071  lecple 16163  joincjn 17152  meetcmee 17153  LSSumclsm 18253  Atomscatm 35045  HLchlt 35132  LHypclh 35766  DVecHcdvh 36860  DIsoBcdib 36920  DIsoCcdic 36954  DIsoHcdih 37010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-riotaBAD 34734
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-iin 4722  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-tpos 7590  df-undef 7637  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-2 11367  df-3 11368  df-4 11369  df-5 11370  df-6 11371  df-n0 11563  df-z 11647  df-uz 11908  df-fz 12553  df-struct 16073  df-ndx 16074  df-slot 16075  df-base 16077  df-sets 16078  df-ress 16079  df-plusg 16169  df-mulr 16170  df-sca 16172  df-vsca 16173  df-0g 16310  df-proset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17796  df-cntz 17954  df-lsm 18255  df-cmn 18399  df-abl 18400  df-mgp 18695  df-ur 18707  df-ring 18754  df-oppr 18828  df-dvdsr 18846  df-unit 18847  df-invr 18877  df-dvr 18888  df-drng 18956  df-lmod 19072  df-lss 19140  df-lsp 19182  df-lvec 19313  df-oposet 34958  df-ol 34960  df-oml 34961  df-covers 35048  df-ats 35049  df-atl 35080  df-cvlat 35104  df-hlat 35133  df-llines 35280  df-lplanes 35281  df-lvols 35282  df-lines 35283  df-psubsp 35285  df-pmap 35286  df-padd 35578  df-lhyp 35770  df-laut 35771  df-ldil 35886  df-ltrn 35887  df-trl 35941  df-tendo 36537  df-edring 36539  df-disoa 36811  df-dvech 36861  df-dib 36921  df-dic 36955  df-dih 37011
This theorem is referenced by:  dihord5apre  37044  dihord  37046
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