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Theorem dihord5b 35360
Description: Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 . (Contributed by NM, 7-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
dihord5b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌))

Proof of Theorem dihord5b
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1057 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1059 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝑌𝐵 ∧ ¬ 𝑌 𝑊))
3 dihord3.b . . . 4 𝐵 = (Base‘𝐾)
4 dihord3.l . . . 4 = (le‘𝐾)
5 eqid 2610 . . . 4 (join‘𝐾) = (join‘𝐾)
6 eqid 2610 . . . 4 (meet‘𝐾) = (meet‘𝐾)
7 eqid 2610 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
8 dihord3.h . . . 4 𝐻 = (LHyp‘𝐾)
93, 4, 5, 6, 7, 8lhpmcvr2 34122 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
101, 2, 9syl2anc 691 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
11 simp1r 1079 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 𝑌)
12 simpl2r 1108 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → 𝑋 𝑊)
13123ad2ant1 1075 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 𝑊)
14 simpl1l 1105 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → 𝐾 ∈ HL)
15143ad2ant1 1075 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝐾 ∈ HL)
16 hllat 33462 . . . . . . . . . . . 12 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1715, 16syl 17 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝐾 ∈ Lat)
18 simpl2l 1107 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → 𝑋𝐵)
19183ad2ant1 1075 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋𝐵)
20 simpl3l 1109 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → 𝑌𝐵)
21203ad2ant1 1075 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑌𝐵)
22 simpl1r 1106 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → 𝑊𝐻)
23223ad2ant1 1075 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑊𝐻)
243, 8lhpbase 34096 . . . . . . . . . . . 12 (𝑊𝐻𝑊𝐵)
2523, 24syl 17 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑊𝐵)
263, 4, 6latlem12 16850 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑊𝐵)) → ((𝑋 𝑌𝑋 𝑊) ↔ 𝑋 (𝑌(meet‘𝐾)𝑊)))
2717, 19, 21, 25, 26syl13anc 1320 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((𝑋 𝑌𝑋 𝑊) ↔ 𝑋 (𝑌(meet‘𝐾)𝑊)))
2811, 13, 27mpbi2and 958 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 (𝑌(meet‘𝐾)𝑊))
29 simp1l1 1147 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 simp1l2 1148 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑋𝐵𝑋 𝑊))
313, 6latmcl 16824 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌(meet‘𝐾)𝑊) ∈ 𝐵)
3217, 21, 25, 31syl3anc 1318 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌(meet‘𝐾)𝑊) ∈ 𝐵)
333, 4, 6latmle2 16849 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌(meet‘𝐾)𝑊) 𝑊)
3417, 21, 25, 33syl3anc 1318 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌(meet‘𝐾)𝑊) 𝑊)
35 eqid 2610 . . . . . . . . . . 11 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
363, 4, 8, 35dibord 35260 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ ((𝑌(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑌(meet‘𝐾)𝑊) 𝑊)) → ((((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ↔ 𝑋 (𝑌(meet‘𝐾)𝑊)))
3729, 30, 32, 34, 36syl112anc 1322 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ↔ 𝑋 (𝑌(meet‘𝐾)𝑊)))
3828, 37mpbird 246 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))
39 eqid 2610 . . . . . . . . . . . 12 ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
408, 39, 29dvhlmod 35211 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((DVecH‘𝐾)‘𝑊) ∈ LMod)
41 eqid 2610 . . . . . . . . . . . 12 (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
4241lsssssubg 18728 . . . . . . . . . . 11 (((DVecH‘𝐾)‘𝑊) ∈ LMod → (LSubSp‘((DVecH‘𝐾)‘𝑊)) ⊆ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
4340, 42syl 17 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (LSubSp‘((DVecH‘𝐾)‘𝑊)) ⊆ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
44 simp2 1055 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊))
45 eqid 2610 . . . . . . . . . . . 12 ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
464, 7, 8, 39, 45, 41diclss 35294 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
4729, 44, 46syl2anc 691 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
4843, 47sseldd 3569 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
493, 4, 8, 39, 35, 41diblss 35271 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑌(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑌(meet‘𝐾)𝑊) 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
5029, 32, 34, 49syl12anc 1316 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
5143, 50sseldd 3569 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
52 eqid 2610 . . . . . . . . . 10 (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
5352lsmub2 17844 . . . . . . . . 9 (((((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)) ∧ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
5448, 51, 53syl2anc 691 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
5538, 54sstrd 3578 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
56 dihord3.i . . . . . . . . 9 𝐼 = ((DIsoH‘𝐾)‘𝑊)
573, 4, 8, 56, 35dihvalb 35338 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
5829, 30, 57syl2anc 691 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
59 simp1l3 1149 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌𝐵 ∧ ¬ 𝑌 𝑊))
60 simp3 1056 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)
613, 4, 5, 6, 7, 8, 56, 35, 45, 39, 52dihvalcq 35337 . . . . . . . 8 (((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))))
6229, 59, 44, 60, 61syl112anc 1322 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (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))))
6355, 58, 623sstr4d 3611 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌))
64633exp 1256 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 𝑊) → ((𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 → (𝐼𝑋) ⊆ (𝐼𝑌))))
6564expd 451 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝑟 ∈ (Atoms‘𝐾) → (¬ 𝑟 𝑊 → ((𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 → (𝐼𝑋) ⊆ (𝐼𝑌)))))
6665imp4a 612 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝑟 ∈ (Atoms‘𝐾) → ((¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌))))
6766rexlimdv 3012 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌)))
6810, 67mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  wss 3540   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  meetcmee 16717  Latclat 16817  SubGrpcsubg 17360  LSSumclsm 17821  LModclmod 18635  LSubSpclss 18702  Atomscatm 33362  HLchlt 33449  LHypclh 34082  DVecHcdvh 35179  DIsoBcdib 35239  DIsoCcdic 35273  DIsoHcdih 35329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-riotaBAD 33051
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-iun 4452  df-iin 4453  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-tpos 7217  df-undef 7264  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-mulr 15731  df-sca 15733  df-vsca 15734  df-0g 15874  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-p1 16812  df-lat 16818  df-clat 16880  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108  df-grp 17197  df-minusg 17198  df-sbg 17199  df-subg 17363  df-cntz 17522  df-lsm 17823  df-cmn 17967  df-abl 17968  df-mgp 18262  df-ur 18274  df-ring 18321  df-oppr 18395  df-dvdsr 18413  df-unit 18414  df-invr 18444  df-dvr 18455  df-drng 18521  df-lmod 18637  df-lss 18703  df-lsp 18742  df-lvec 18873  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-llines 33596  df-lplanes 33597  df-lvols 33598  df-lines 33599  df-psubsp 33601  df-pmap 33602  df-padd 33894  df-lhyp 34086  df-laut 34087  df-ldil 34202  df-ltrn 34203  df-trl 34258  df-tendo 34855  df-edring 34857  df-disoa 35130  df-dvech 35180  df-dib 35240  df-dic 35274  df-dih 35330
This theorem is referenced by:  dihord  35365
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