Step |
Hyp |
Ref |
Expression |
1 |
|
tendoset.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
tendoset.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
tendoset.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendoset.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
tendoset.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
1 2
|
tendofset |
⊢ ( 𝐾 ∈ 𝑉 → ( TEndo ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝐾 ∈ 𝑉 → ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) ‘ 𝑊 ) ) |
8 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
9 |
8 8
|
feq23d |
⊢ ( 𝑤 = 𝑊 → ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↔ 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
10 |
8
|
raleqdv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ↔ ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ) ) |
11 |
8 10
|
raleqbidv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ↔ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ) |
13 |
12 4
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) = 𝑅 ) |
14 |
13
|
fveq1d |
⊢ ( 𝑤 = 𝑊 → ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) = ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ) |
15 |
13
|
fveq1d |
⊢ ( 𝑤 = 𝑊 → ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) = ( 𝑅 ‘ 𝑓 ) ) |
16 |
14 15
|
breq12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ↔ ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) |
17 |
8 16
|
raleqbidv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ↔ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) |
18 |
9 11 17
|
3anbi123d |
⊢ ( 𝑤 = 𝑊 → ( ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) ↔ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) ) |
19 |
18
|
abbidv |
⊢ ( 𝑤 = 𝑊 → { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } = { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ) |
20 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) |
21 |
|
fvex |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
22 |
21 21
|
mapval |
⊢ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↑m ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) = { 𝑠 ∣ 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) } |
23 |
|
ovex |
⊢ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↑m ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ V |
24 |
22 23
|
eqeltrri |
⊢ { 𝑠 ∣ 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) } ∈ V |
25 |
|
simp1 |
⊢ ( ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) → 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
26 |
25
|
ss2abi |
⊢ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ⊆ { 𝑠 ∣ 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) } |
27 |
24 26
|
ssexi |
⊢ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ∈ V |
28 |
19 20 27
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) ‘ 𝑊 ) = { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ) |
29 |
3 3
|
feq23i |
⊢ ( 𝑠 : 𝑇 ⟶ 𝑇 ↔ 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
30 |
3
|
raleqi |
⊢ ( ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ↔ ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ) |
31 |
3 30
|
raleqbii |
⊢ ( ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ↔ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ) |
32 |
3
|
raleqi |
⊢ ( ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ↔ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) |
33 |
29 31 32
|
3anbi123i |
⊢ ( ( 𝑠 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ↔ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) |
34 |
33
|
abbii |
⊢ { 𝑠 ∣ ( 𝑠 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } = { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } |
35 |
28 34
|
eqtr4di |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) ‘ 𝑊 ) = { 𝑠 ∣ ( 𝑠 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ) |
36 |
7 35
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = { 𝑠 ∣ ( 𝑠 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ) |
37 |
5 36
|
syl5eq |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐸 = { 𝑠 ∣ ( 𝑠 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) } ) |