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 ) |
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 ) |