Metamath Proof Explorer

Theorem djhjlj

Description: DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014)

Ref Expression
Hypotheses djhj.k ⊢ ∨ = ( join ‘ 𝐾 )
djhj.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
djhj.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
djhj.j ⊢ 𝐽 = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 )
djhj.w ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
djhj.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
djhj.y ⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
Assertion djhjlj ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 djhj.k ⊢ ∨ = ( join ‘ 𝐾 )
2 djhj.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 djhj.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 djhj.j ⊢ 𝐽 = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 )
5 djhj.w ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 djhj.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
7 djhj.y ⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
8 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9 8 2 3 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
10 5 6 9 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
11 8 2 3 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) )
12 5 7 11 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) )
13 8 1 2 3 4 djhlj ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) 𝐽 ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ) )
14 5 10 12 13 syl12anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) 𝐽 ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ) )
15 2 3 dihcnvid2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
16 5 6 15 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
17 2 3 dihcnvid2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) = 𝑌 )
18 5 7 17 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) = 𝑌 )
19 16 18 oveq12d ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) 𝐽 ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ) = ( 𝑋 𝐽 𝑌 ) )
20 14 19 eqtr2d ⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) )