Step |
Hyp |
Ref |
Expression |
1 |
|
ltrncvr.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ltrncvr.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
3 |
|
ltrncvr.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrncvr.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ 𝑉 ) |
6 |
|
eqid |
⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 ) |
7 |
3 6 4
|
ltrnlaut |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) ) |
8 |
7
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) ) |
9 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
10 |
|
simp3r |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
11 |
1 2 6
|
lautcvr |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ ( 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) 𝐶 ( 𝐹 ‘ 𝑌 ) ) ) |
12 |
5 8 9 10 11
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) 𝐶 ( 𝐹 ‘ 𝑌 ) ) ) |