Step |
Hyp |
Ref |
Expression |
1 |
|
tendo0.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tendo0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
tendo0.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendo0.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
tendo0.o |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
6 |
|
tendo0pl.p |
⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) |
7 |
1 2 3 4 5
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 ) |
8 |
7
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑂 ∈ 𝐸 ) |
9 |
2 3 4 6
|
tendoplcom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝑂 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = ( 𝑂 𝑃 𝑆 ) ) |
10 |
8 9
|
mpd3an3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = ( 𝑂 𝑃 𝑆 ) ) |
11 |
1 2 3 4 5 6
|
tendo0pl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑆 ) = 𝑆 ) |
12 |
10 11
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = 𝑆 ) |