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 ) ) )

Metamath Proof Explorer

Theorem dihjatcc

Proof