Metamath Proof Explorer
Description: In dimension zero, any two pairs of points are congruent.
(Contributed by Thierry Arnoux, 12-Apr-2019)
|
|
Ref |
Expression |
|
Hypotheses |
tgbtwndiff.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
|
|
tgbtwndiff.d |
⊢ − = ( dist ‘ 𝐺 ) |
|
|
tgbtwndiff.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
|
|
tgbtwndiff.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
|
|
tgbtwndiff.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
|
|
tgbtwndiff.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
|
|
tgldim0itv.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
|
|
tgldim0itv.p |
⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = 1 ) |
|
|
tgldim0itv.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
|
Assertion |
tgldim0cgr |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
tgbtwndiff.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
tgbtwndiff.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
tgbtwndiff.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
tgbtwndiff.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
tgbtwndiff.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
tgbtwndiff.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
tgldim0itv.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
8 |
|
tgldim0itv.p |
⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = 1 ) |
9 |
|
tgldim0itv.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
10 |
1 8 5 7
|
tgldim0eq |
⊢ ( 𝜑 → 𝐴 = 𝐶 ) |
11 |
1 8 6 9
|
tgldim0eq |
⊢ ( 𝜑 → 𝐵 = 𝐷 ) |
12 |
10 11
|
oveq12d |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) ) |