Theorem diael

Description: A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014)

	Ref	Expression
Hypotheses	diass.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	diass.l	⊢ ≤ = ( le ‘ 𝐾 )
	diass.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	diass.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	diass.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	diael	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝐹 ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		diass.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		diass.l	⊢ ≤ = ( le ‘ 𝐾 )
3		diass.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		diass.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		diass.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
6	1 2 3 4 5	diass	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 )
7	6	sseld	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) → 𝐹 ∈ 𝑇 ) )
8	7	3impia	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝐹 ∈ 𝑇 )