Step |
Hyp |
Ref |
Expression |
1 |
|
dvhop.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dvhop.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dvhop.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dvhop.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dvhop.p |
⊢ 𝑃 = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑐 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑐 ) ∘ ( 𝑏 ‘ 𝑐 ) ) ) ) |
6 |
|
dvhop.a |
⊢ 𝐴 = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ( 2nd ‘ 𝑓 ) 𝑃 ( 2nd ‘ 𝑔 ) ) 〉 ) |
7 |
|
dvhop.s |
⊢ 𝑆 = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) |
8 |
|
dvhop.o |
⊢ 𝑂 = ( 𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
9 |
|
simprr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 𝑈 ∈ 𝐸 ) |
10 |
1 2 3
|
idltrn |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ 𝑇 ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( I ↾ 𝐵 ) ∈ 𝑇 ) |
12 |
2 3 4
|
tendoidcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝑇 ) ∈ 𝐸 ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( I ↾ 𝑇 ) ∈ 𝐸 ) |
14 |
7
|
dvhopspN |
⊢ ( ( 𝑈 ∈ 𝐸 ∧ ( ( I ↾ 𝐵 ) ∈ 𝑇 ∧ ( I ↾ 𝑇 ) ∈ 𝐸 ) ) → ( 𝑈 𝑆 〈 ( I ↾ 𝐵 ) , ( I ↾ 𝑇 ) 〉 ) = 〈 ( 𝑈 ‘ ( I ↾ 𝐵 ) ) , ( 𝑈 ∘ ( I ↾ 𝑇 ) ) 〉 ) |
15 |
9 11 13 14
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 𝑈 𝑆 〈 ( I ↾ 𝐵 ) , ( I ↾ 𝑇 ) 〉 ) = 〈 ( 𝑈 ‘ ( I ↾ 𝐵 ) ) , ( 𝑈 ∘ ( I ↾ 𝑇 ) ) 〉 ) |
16 |
1 2 4
|
tendoid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → ( 𝑈 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
17 |
16
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 𝑈 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
18 |
2 3 4
|
tendo1mulr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → ( 𝑈 ∘ ( I ↾ 𝑇 ) ) = 𝑈 ) |
19 |
18
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 𝑈 ∘ ( I ↾ 𝑇 ) ) = 𝑈 ) |
20 |
17 19
|
opeq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 〈 ( 𝑈 ‘ ( I ↾ 𝐵 ) ) , ( 𝑈 ∘ ( I ↾ 𝑇 ) ) 〉 = 〈 ( I ↾ 𝐵 ) , 𝑈 〉 ) |
21 |
15 20
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 𝑈 𝑆 〈 ( I ↾ 𝐵 ) , ( I ↾ 𝑇 ) 〉 ) = 〈 ( I ↾ 𝐵 ) , 𝑈 〉 ) |
22 |
21
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 〈 𝐹 , 𝑂 〉 𝐴 ( 𝑈 𝑆 〈 ( I ↾ 𝐵 ) , ( I ↾ 𝑇 ) 〉 ) ) = ( 〈 𝐹 , 𝑂 〉 𝐴 〈 ( I ↾ 𝐵 ) , 𝑈 〉 ) ) |
23 |
|
simprl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 𝐹 ∈ 𝑇 ) |
24 |
1 2 3 4 8
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 𝑂 ∈ 𝐸 ) |
26 |
6
|
dvhopaddN |
⊢ ( ( ( 𝐹 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸 ) ∧ ( ( I ↾ 𝐵 ) ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 〈 𝐹 , 𝑂 〉 𝐴 〈 ( I ↾ 𝐵 ) , 𝑈 〉 ) = 〈 ( 𝐹 ∘ ( I ↾ 𝐵 ) ) , ( 𝑂 𝑃 𝑈 ) 〉 ) |
27 |
23 25 11 9 26
|
syl22anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 〈 𝐹 , 𝑂 〉 𝐴 〈 ( I ↾ 𝐵 ) , 𝑈 〉 ) = 〈 ( 𝐹 ∘ ( I ↾ 𝐵 ) ) , ( 𝑂 𝑃 𝑈 ) 〉 ) |
28 |
1 2 3
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
29 |
28
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
30 |
|
f1of |
⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 ⟶ 𝐵 ) |
31 |
|
fcoi1 |
⊢ ( 𝐹 : 𝐵 ⟶ 𝐵 → ( 𝐹 ∘ ( I ↾ 𝐵 ) ) = 𝐹 ) |
32 |
29 30 31
|
3syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 𝐹 ∘ ( I ↾ 𝐵 ) ) = 𝐹 ) |
33 |
1 2 3 4 8 5
|
tendo0pl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑈 ) = 𝑈 ) |
34 |
33
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → ( 𝑂 𝑃 𝑈 ) = 𝑈 ) |
35 |
32 34
|
opeq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 〈 ( 𝐹 ∘ ( I ↾ 𝐵 ) ) , ( 𝑂 𝑃 𝑈 ) 〉 = 〈 𝐹 , 𝑈 〉 ) |
36 |
22 27 35
|
3eqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸 ) ) → 〈 𝐹 , 𝑈 〉 = ( 〈 𝐹 , 𝑂 〉 𝐴 ( 𝑈 𝑆 〈 ( I ↾ 𝐵 ) , ( I ↾ 𝑇 ) 〉 ) ) ) |