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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( ( ( 𝐼𝑝 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) ∩ ( 𝐼𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( ( 𝐼𝑝 ) ∩ ( 𝐼𝑌 ) ) ) )

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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
11 3 7 10 dvhlmod ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝑈 ∈ LMod )
12 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
13 12 lsssssubg ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
14 11 13 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
15 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝐾 ∈ HL )
16 15 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝐾 ∈ Lat )
17 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝑋𝐵 )
18 simp2r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝑌𝐵 )
19 1 5 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) ∈ 𝐵 )
20 16 17 18 19 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝑋 𝑌 ) ∈ 𝐵 )
21 1 3 9 7 12 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋 𝑌 ) ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
22 10 20 21 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
23 14 22 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
24 1 6 atbase ( 𝑝𝐴𝑝𝐵 )
25 24 3ad2ant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝑝𝐵 )
26 1 3 9 7 12 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑝𝐵 ) → ( 𝐼𝑝 ) ∈ ( LSubSp ‘ 𝑈 ) )
27 10 25 26 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼𝑝 ) ∈ ( LSubSp ‘ 𝑈 ) )
28 14 27 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) )
29 1 3 9 7 12 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ) → ( 𝐼𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
30 10 18 29 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
31 14 30 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) )
32 1 2 5 latmle2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) 𝑌 )
33 16 17 18 32 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝑋 𝑌 ) 𝑌 )
34 1 2 3 9 dihord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋 𝑌 ) ∈ 𝐵𝑌𝐵 ) → ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ⊆ ( 𝐼𝑌 ) ↔ ( 𝑋 𝑌 ) 𝑌 ) )
35 10 20 18 34 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ⊆ ( 𝐼𝑌 ) ↔ ( 𝑋 𝑌 ) 𝑌 ) )
36 33 35 mpbird ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ⊆ ( 𝐼𝑌 ) )
37 8 lsmmod ( ( ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) ∧ ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ⊆ ( 𝐼𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( ( 𝐼𝑝 ) ∩ ( 𝐼𝑌 ) ) ) = ( ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( 𝐼𝑝 ) ) ∩ ( 𝐼𝑌 ) ) )
38 23 28 31 36 37 syl31anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( ( 𝐼𝑝 ) ∩ ( 𝐼𝑌 ) ) ) = ( ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( 𝐼𝑝 ) ) ∩ ( 𝐼𝑌 ) ) )
39 lmodabl ( 𝑈 ∈ LMod → 𝑈 ∈ Abel )
40 11 39 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → 𝑈 ∈ Abel )
41 8 lsmcom ( ( 𝑈 ∈ Abel ∧ ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( 𝐼𝑝 ) ) = ( ( 𝐼𝑝 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) )
42 40 23 28 41 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( 𝐼𝑝 ) ) = ( ( 𝐼𝑝 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) )
43 42 ineq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( 𝐼𝑝 ) ) ∩ ( 𝐼𝑌 ) ) = ( ( ( 𝐼𝑝 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) ∩ ( 𝐼𝑌 ) ) )
44 38 43 eqtr2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑝𝐴 ) → ( ( ( 𝐼𝑝 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) ∩ ( 𝐼𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ( ( 𝐼𝑝 ) ∩ ( 𝐼𝑌 ) ) ) )