Step |
Hyp |
Ref |
Expression |
1 |
|
trgcgrg.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
trgcgrg.m |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
trgcgrg.r |
⊢ ∼ = ( cgrG ‘ 𝐺 ) |
4 |
|
trgcgrg.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
trgcgrg.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
trgcgrg.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
trgcgrg.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
8 |
|
trgcgrg.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
9 |
|
trgcgrg.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) |
10 |
|
trgcgrg.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) |
11 |
|
trgcgr.1 |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ) |
12 |
|
trgcgr.2 |
⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ) |
13 |
|
trgcgr.3 |
⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) |
14 |
1 2 3 4 5 6 7 8 9 10
|
trgcgrg |
⊢ ( 𝜑 → ( 〈“ 𝐴 𝐵 𝐶 ”〉 ∼ 〈“ 𝐷 𝐸 𝐹 ”〉 ↔ ( ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ∧ ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ∧ ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) ) ) |
15 |
11 12 13 14
|
mpbir3and |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ∼ 〈“ 𝐷 𝐸 𝐹 ”〉 ) |