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Metamath Proof Explorer

Theorem dihjat6

Description: Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015)

Ref Expression
Hypotheses dihjat6.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjat6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjat6.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihjat6.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjat6.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjat6.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
dihjat6.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihjat6.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
dihjat6.q ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
Assertion dihjat6 ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑋 ⊕ 𝑄 ) ) = ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 dihjat6.j ⊢ ∨ = ( join ‘ 𝐾 )
2 dihjat6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dihjat6.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 dihjat6.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dihjat6.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
6 dihjat6.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
7 dihjat6.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8 dihjat6.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
9 dihjat6.q ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
10 1 2 3 4 5 6 7 8 9 dihjat4 ⊢ ( 𝜑 → ( 𝑋 ⊕ 𝑄 ) = ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ) )
11 10 fveq2d ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑋 ⊕ 𝑄 ) ) = ( ◡ 𝐼 ‘ ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ) ) )
12 7 simpld ⊢ ( 𝜑 → 𝐾 ∈ HL )
13 12 hllatd ⊢ ( 𝜑 → 𝐾 ∈ Lat )
14 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
15 14 2 3 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
16 7 8 15 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
17 2 4 3 6 dih1dimat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ ran 𝐼 )
18 7 9 17 syl2anc ⊢ ( 𝜑 → 𝑄 ∈ ran 𝐼 )
19 14 2 3 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
20 7 18 19 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
21 14 1 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
22 13 16 20 21 syl3anc ⊢ ( 𝜑 → ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
23 14 2 3 dihcnvid1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ) ) = ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) )
24 7 22 23 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) ) ) = ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) )
25 11 24 eqtrd ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑋 ⊕ 𝑄 ) ) = ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑄 ) ) )