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 ∧ 𝑊𝐻 ) )
dihjatcc.p ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
dihjatcc.q ( 𝜑 → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
Assertion dihjatcc ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) )

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 ∧ 𝑊𝐻 ) )
9 dihjatcc.p ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
10 dihjatcc.q ( 𝜑 → ( 𝑄𝐴 ∧ ¬ 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 ) ↦ ( 𝑎𝑑 ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑎𝑑 ) ) )
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 ) ↦ ( ( 𝑎𝑑 ) ∘ ( 𝑏𝑑 ) ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎𝑑 ) ∘ ( 𝑏𝑑 ) ) ) )
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 ) ) ⊆ ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) )
24 11 1 2 3 12 4 5 6 7 13 8 9 10 23 dihjatcclem2 ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) )