Step |
Hyp |
Ref |
Expression |
1 |
|
mirval.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
mirval.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
mirval.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
mirval.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
5 |
|
mirval.s |
⊢ 𝑆 = ( pInvG ‘ 𝐺 ) |
6 |
|
mirval.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
7 |
|
miduniq1.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
8 |
|
miduniq1.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
9 |
|
miduniq1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑃 ) |
10 |
|
miduniq1.e |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐵 ) ‘ 𝑋 ) ) |
11 |
|
eqid |
⊢ ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐴 ) |
12 |
1 2 3 4 5 6 7 11 9
|
mircl |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) ∈ 𝑃 ) |
13 |
|
eqidd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) ) |
14 |
10
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) ) |
15 |
1 2 3 4 5 6 7 8 9 12 13 14
|
miduniq |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |