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 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
1 2 3 4 5
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 ) |
9 |
8
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑂 ∈ 𝐸 ) |
10 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑆 ∈ 𝐸 ) |
11 |
2 3 4 6
|
tendoplcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑂 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑆 ) ∈ 𝐸 ) |
12 |
7 9 10 11
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑆 ) ∈ 𝐸 ) |
13 |
|
simpll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
13 8
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → 𝑂 ∈ 𝐸 ) |
15 |
|
simplr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → 𝑆 ∈ 𝐸 ) |
16 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → 𝑔 ∈ 𝑇 ) |
17 |
6 3
|
tendopl2 |
⊢ ( ( 𝑂 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑂 𝑃 𝑆 ) ‘ 𝑔 ) = ( ( 𝑂 ‘ 𝑔 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ) |
18 |
14 15 16 17
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑂 𝑃 𝑆 ) ‘ 𝑔 ) = ( ( 𝑂 ‘ 𝑔 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ) |
19 |
5 1
|
tendo02 |
⊢ ( 𝑔 ∈ 𝑇 → ( 𝑂 ‘ 𝑔 ) = ( I ↾ 𝐵 ) ) |
20 |
19
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( 𝑂 ‘ 𝑔 ) = ( I ↾ 𝐵 ) ) |
21 |
20
|
coeq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑂 ‘ 𝑔 ) ∘ ( 𝑆 ‘ 𝑔 ) ) = ( ( I ↾ 𝐵 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ) |
22 |
2 3 4
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( 𝑆 ‘ 𝑔 ) ∈ 𝑇 ) |
23 |
22
|
3expa |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( 𝑆 ‘ 𝑔 ) ∈ 𝑇 ) |
24 |
1 2 3
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝑔 ) ∈ 𝑇 ) → ( 𝑆 ‘ 𝑔 ) : 𝐵 –1-1-onto→ 𝐵 ) |
25 |
13 23 24
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( 𝑆 ‘ 𝑔 ) : 𝐵 –1-1-onto→ 𝐵 ) |
26 |
|
f1of |
⊢ ( ( 𝑆 ‘ 𝑔 ) : 𝐵 –1-1-onto→ 𝐵 → ( 𝑆 ‘ 𝑔 ) : 𝐵 ⟶ 𝐵 ) |
27 |
|
fcoi2 |
⊢ ( ( 𝑆 ‘ 𝑔 ) : 𝐵 ⟶ 𝐵 → ( ( I ↾ 𝐵 ) ∘ ( 𝑆 ‘ 𝑔 ) ) = ( 𝑆 ‘ 𝑔 ) ) |
28 |
25 26 27
|
3syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( ( I ↾ 𝐵 ) ∘ ( 𝑆 ‘ 𝑔 ) ) = ( 𝑆 ‘ 𝑔 ) ) |
29 |
18 21 28
|
3eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑂 𝑃 𝑆 ) ‘ 𝑔 ) = ( 𝑆 ‘ 𝑔 ) ) |
30 |
29
|
ralrimiva |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ∀ 𝑔 ∈ 𝑇 ( ( 𝑂 𝑃 𝑆 ) ‘ 𝑔 ) = ( 𝑆 ‘ 𝑔 ) ) |
31 |
2 3 4
|
tendoeq1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑂 𝑃 𝑆 ) ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ∧ ∀ 𝑔 ∈ 𝑇 ( ( 𝑂 𝑃 𝑆 ) ‘ 𝑔 ) = ( 𝑆 ‘ 𝑔 ) ) → ( 𝑂 𝑃 𝑆 ) = 𝑆 ) |
32 |
7 12 10 30 31
|
syl121anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑆 ) = 𝑆 ) |