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Metamath Proof Explorer

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 ) )