Metamath Proof Explorer


Theorem dihmeetlem11N

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 dihmeetlem11N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) = ( ( 𝐼𝑋 ) ∩ ( 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 1 2 3 4 5 6 7 8 9 dihmeetlem10N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑝 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ) )
11 10 ineq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) = ( ( ( 𝐼𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ) ∩ ( 𝐼𝑌 ) ) )
12 inass ( ( ( 𝐼𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ) ∩ ( 𝐼𝑌 ) ) = ( ( 𝐼𝑋 ) ∩ ( ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) )
13 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → 𝐾 ∈ HL )
14 13 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → 𝐾 ∈ Lat )
15 simpl3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → 𝑌𝐵 )
16 simprll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → 𝑝𝐴 )
17 1 6 atbase ( 𝑝𝐴𝑝𝐵 )
18 16 17 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → 𝑝𝐵 )
19 1 2 4 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑝𝐵 ) → 𝑌 ( 𝑌 𝑝 ) )
20 14 15 18 19 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → 𝑌 ( 𝑌 𝑝 ) )
21 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
22 1 4 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑝𝐵 ) → ( 𝑌 𝑝 ) ∈ 𝐵 )
23 14 15 18 22 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( 𝑌 𝑝 ) ∈ 𝐵 )
24 1 2 3 9 dihord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑌 𝑝 ) ∈ 𝐵 ) → ( ( 𝐼𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ↔ 𝑌 ( 𝑌 𝑝 ) ) )
25 21 15 23 24 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( 𝐼𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ↔ 𝑌 ( 𝑌 𝑝 ) ) )
26 20 25 mpbird ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( 𝐼𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) )
27 sseqin2 ( ( 𝐼𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ↔ ( ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) = ( 𝐼𝑌 ) )
28 26 27 sylib ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) = ( 𝐼𝑌 ) )
29 28 ineq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( 𝐼𝑋 ) ∩ ( ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑌 ) ) )
30 12 29 syl5eq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( ( 𝐼𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ) ∩ ( 𝐼𝑌 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑌 ) ) )
31 11 30 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ 𝑝 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑝 ) ) ∩ ( 𝐼𝑌 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑌 ) ) )