Metamath Proof Explorer

Theorem dihmeetlem9N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem9.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem9.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem9.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetlem9.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem9.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem9.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihmeetlem9.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem9.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihmeetlem9.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihmeetlem9N ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem9.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem9.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem9.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihmeetlem9.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihmeetlem9.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihmeetlem9.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihmeetlem9.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihmeetlem9.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihmeetlem9.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 3 7 10 dvhlmod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑈 ∈ LMod )
12 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
13 12 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
14 11 13 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
15 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ HL )
16 15 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ Lat )
17 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
18 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 )
19 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
20 16 17 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
21 1 3 9 7 12 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
22 10 20 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
23 14 22 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
24 1 6 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
25 24 3ad2ant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ∈ 𝐵 )
26 1 3 9 7 12 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑝 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑝 ) ∈ ( LSubSp ‘ 𝑈 ) )
27 10 25 26 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑝 ) ∈ ( LSubSp ‘ 𝑈 ) )
28 14 27 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) )
29 1 3 9 7 12 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
30 10 18 29 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
31 14 30 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) )
32 1 2 5 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 )
33 16 17 18 32 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 )
34 1 2 3 9 dihord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) )
35 10 20 18 34 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) )
36 33 35 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) )
37 8 lsmmod ⊢ ( ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
38 23 28 31 36 37 syl31anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
39 lmodabl ⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Abel )
40 11 39 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑈 ∈ Abel )
41 8 lsmcom ⊢ ( ( 𝑈 ∈ Abel ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )
42 40 23 28 41 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )
43 42 ineq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
44 38 43 eqtr2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) )