Theorem dihssxp

Description: An isomorphism H value is included in the vector space (expressed as T X. E ). (Contributed by NM, 26-Sep-2014)

	Ref	Expression
Hypotheses	dihssxp.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihssxp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihssxp.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dihssxp.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dihssxp.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihssxp.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dihssxp.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	dihssxp	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝑇 × 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		dihssxp.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihssxp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dihssxp.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		dihssxp.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5		dihssxp.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
6		dihssxp.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		dihssxp.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
9		eqid	⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
10	1 2 5 8 9	dihss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
11	6 7 10	syl2anc	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ⊆ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
12	2 3 4 8 9	dvhvbase	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑇 × 𝐸 ) )
13	6 12	syl	⊢ ( 𝜑 → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑇 × 𝐸 ) )
14	11 13	sseqtrd	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝑇 × 𝐸 ) )