Metamath Proof Explorer

Theorem dihjatcc

Description: Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014)

Ref Expression
Hypotheses dihjatcc.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjatcc.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjatcc.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjatcc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjatcc.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjatcc.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjatcc.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihjatcc.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihjatcc.p ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
dihjatcc.q ⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
Assertion dihjatcc ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 dihjatcc.l ⊢ ≤ = ( le ‘ 𝐾 )
2 dihjatcc.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dihjatcc.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihjatcc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dihjatcc.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 dihjatcc.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
7 dihjatcc.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjatcc.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 dihjatcc.p ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
10 dihjatcc.q ⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
11 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
13 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 )
14 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
15 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
16 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
17 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
18 eqid ⊢ ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑃 ) = ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑃 )
19 eqid ⊢ ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 )
20 eqid ⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑑 ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑑 ) ) )
21 eqid ⊢ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) )
22 eqid ⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎 ‘ 𝑑 ) ∘ ( 𝑏 ‘ 𝑑 ) ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎 ‘ 𝑑 ) ∘ ( 𝑏 ‘ 𝑑 ) ) ) )
23 11 1 2 3 12 4 5 6 7 13 8 9 10 14 15 16 17 18 19 20 21 22 dihjatcclem4 ⊢ ( 𝜑 → ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ⊆ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
24 11 1 2 3 12 4 5 6 7 13 8 9 10 23 dihjatcclem2 ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )