Theorem dihopcl

Description: Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014)

	Ref	Expression
Hypotheses	dihopcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihopcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihopcl.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dihopcl.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dihopcl.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihopcl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dihopcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	dihopcl.y	⊢ ( 𝜑 → ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) )
Assertion	dihopcl	⊢ ( 𝜑 → ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		dihopcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihopcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dihopcl.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		dihopcl.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5		dihopcl.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
6		dihopcl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		dihopcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		dihopcl.y	⊢ ( 𝜑 → ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) )
9	1 2 3 4 5 6 7	dihssxp	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝑇 × 𝐸 ) )
10	9 8	sseldd	⊢ ( 𝜑 → ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝑇 × 𝐸 ) )
11		opelxp	⊢ ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝑇 × 𝐸 ) ↔ ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) )
12	10 11	sylib	⊢ ( 𝜑 → ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) )