Theorem diass

Description: The value of the partial isomorphism A is a set of translations, i.e., a set of vectors. (Contributed by NM, 26-Nov-2013)

	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	diass	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ 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		eqid	⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 4 6 5	diaval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑋 } )
8		ssrab2	⊢ { 𝑓 ∈ 𝑇 ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑋 } ⊆ 𝑇
9	7 8	eqsstrdi	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 )