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 3 4 5
|
istendo |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) ) |
7 |
|
2fveq3 |
⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) = ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ) |
8 |
|
fveq2 |
⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) ) |
9 |
7 8
|
breq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ↔ ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) |
10 |
9
|
rspccv |
⊢ ( ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) |
11 |
10
|
3ad2ant3 |
⊢ ( ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) |
12 |
6 11
|
syl6bi |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) ) |
13 |
12
|
3imp |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |