Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemn8.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemn8.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemn8.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
cdlemn8.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
cdlemn8.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
cdlemn8.o |
⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
7 |
|
cdlemn8.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
cdlemn8.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
cdlemn8.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
cdlemn8.s |
⊢ + = ( +g ‘ 𝑈 ) |
11 |
|
cdlemn8.f |
⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 ) |
12 |
|
cdlemn8.g |
⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑅 ) |
13 |
|
coass |
⊢ ( ( ◡ 𝐹 ∘ 𝐹 ) ∘ 𝑔 ) = ( ◡ 𝐹 ∘ ( 𝐹 ∘ 𝑔 ) ) |
14 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
2 3 4 5
|
lhpocnel2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
16 |
15
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
17 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
18 |
2 3 4 7 11
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 ) |
19 |
14 16 17 18
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝐹 ∈ 𝑇 ) |
20 |
1 4 7
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
21 |
14 19 20
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
22 |
|
f1ococnv1 |
⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐵 ) ) |
23 |
21 22
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐵 ) ) |
24 |
23
|
coeq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( ◡ 𝐹 ∘ 𝐹 ) ∘ 𝑔 ) = ( ( I ↾ 𝐵 ) ∘ 𝑔 ) ) |
25 |
|
simp32 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝑔 ∈ 𝑇 ) |
26 |
1 4 7
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ) → 𝑔 : 𝐵 –1-1-onto→ 𝐵 ) |
27 |
14 25 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝑔 : 𝐵 –1-1-onto→ 𝐵 ) |
28 |
|
f1of |
⊢ ( 𝑔 : 𝐵 –1-1-onto→ 𝐵 → 𝑔 : 𝐵 ⟶ 𝐵 ) |
29 |
|
fcoi2 |
⊢ ( 𝑔 : 𝐵 ⟶ 𝐵 → ( ( I ↾ 𝐵 ) ∘ 𝑔 ) = 𝑔 ) |
30 |
27 28 29
|
3syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( I ↾ 𝐵 ) ∘ 𝑔 ) = 𝑔 ) |
31 |
24 30
|
eqtr2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝑔 = ( ( ◡ 𝐹 ∘ 𝐹 ) ∘ 𝑔 ) ) |
32 |
1 2 3 4 5 6 7 8 9 10 11 12
|
cdlemn7 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝐺 = ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) ∧ ( I ↾ 𝑇 ) = 𝑠 ) ) |
33 |
32
|
simpld |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝐺 = ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) ) |
34 |
32
|
simprd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( I ↾ 𝑇 ) = 𝑠 ) |
35 |
34
|
fveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( I ↾ 𝑇 ) ‘ 𝐹 ) = ( 𝑠 ‘ 𝐹 ) ) |
36 |
|
fvresi |
⊢ ( 𝐹 ∈ 𝑇 → ( ( I ↾ 𝑇 ) ‘ 𝐹 ) = 𝐹 ) |
37 |
19 36
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( I ↾ 𝑇 ) ‘ 𝐹 ) = 𝐹 ) |
38 |
35 37
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑠 ‘ 𝐹 ) = 𝐹 ) |
39 |
38
|
coeq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) = ( 𝐹 ∘ 𝑔 ) ) |
40 |
33 39
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝐺 = ( 𝐹 ∘ 𝑔 ) ) |
41 |
40
|
coeq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ◡ 𝐹 ∘ 𝐺 ) = ( ◡ 𝐹 ∘ ( 𝐹 ∘ 𝑔 ) ) ) |
42 |
13 31 41
|
3eqtr4a |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝑔 = ( ◡ 𝐹 ∘ 𝐺 ) ) |
43 |
4 7
|
ltrncnv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 ) |
44 |
14 19 43
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ◡ 𝐹 ∈ 𝑇 ) |
45 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) |
46 |
2 3 4 7 12
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 ) |
47 |
14 16 45 46
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝐺 ∈ 𝑇 ) |
48 |
4 7
|
ltrncom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ◡ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ◡ 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ ◡ 𝐹 ) ) |
49 |
14 44 47 48
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ◡ 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ ◡ 𝐹 ) ) |
50 |
42 49
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 ( 𝑠 ‘ 𝐹 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → 𝑔 = ( 𝐺 ∘ ◡ 𝐹 ) ) |