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 ) } ) |