Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnel.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrnel.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrnel.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrnel.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝐺 ∈ 𝑇 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
7 3 4
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) |
9 |
5 6 8
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) |
10 |
|
f1of |
⊢ ( 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ) |
11 |
9 10
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ) |
12 |
7 2
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
13 |
12
|
3ad2ant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
14 |
|
fvco3 |
⊢ ( ( 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) |
15 |
11 13 14
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) |