Theorem dia1dim

Description: Two expressions for the 1-dimensional subspaces of partial vector space A (when F is a nonzero vector i.e. non-identity translation). Remark after Lemma L in Crawley p. 120 line 21. (Contributed by NM, 15-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dia1dim.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dia1dim.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dia1dim.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	dia1dim.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dia1dim.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dia1dim	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑔 ∣ ∃ 𝑠 ∈ 𝐸 𝑔 = ( 𝑠 ‘ 𝐹 ) } )

Proof

Step	Hyp	Ref	Expression
1		dia1dim.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dia1dim.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dia1dim.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
4		dia1dim.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5		dia1dim.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
6		simpl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8	7 1 2 3	trlcl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
9		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
10	9 1 2 3	trlle	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ( le ‘ 𝐾 ) 𝑊 )
11	7 9 1 2 3 5	diaval	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑔 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑔 ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) } )
12	6 8 10 11	syl12anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑔 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑔 ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) } )
13	9 1 2 3 4	dva1dim	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → { 𝑔 ∣ ∃ 𝑠 ∈ 𝐸 𝑔 = ( 𝑠 ‘ 𝐹 ) } = { 𝑔 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑔 ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) } )
14	12 13	eqtr4d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑔 ∣ ∃ 𝑠 ∈ 𝐸 𝑔 = ( 𝑠 ‘ 𝐹 ) } )