Step |
Hyp |
Ref |
Expression |
1 |
|
ismid.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
ismid.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
ismid.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
ismid.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
ismid.1 |
⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 ) |
6 |
|
lmif.m |
⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 ) |
7 |
|
lmif.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
8 |
|
lmif.d |
⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 ) |
9 |
|
lmicl.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
10 |
|
lmieq.c |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
11 |
|
lmieq.d |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ 𝐵 ) ) |
12 |
|
fveqeq2 |
⊢ ( 𝑏 = 𝐴 → ( ( 𝑀 ‘ 𝑏 ) = ( 𝑀 ‘ 𝐵 ) ↔ ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ 𝐵 ) ) ) |
13 |
|
fveqeq2 |
⊢ ( 𝑏 = 𝐵 → ( ( 𝑀 ‘ 𝑏 ) = ( 𝑀 ‘ 𝐵 ) ↔ ( 𝑀 ‘ 𝐵 ) = ( 𝑀 ‘ 𝐵 ) ) ) |
14 |
1 2 3 4 5 6 7 8 10
|
lmicl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ 𝑃 ) |
15 |
1 2 3 4 5 6 7 8 14
|
lmireu |
⊢ ( 𝜑 → ∃! 𝑏 ∈ 𝑃 ( 𝑀 ‘ 𝑏 ) = ( 𝑀 ‘ 𝐵 ) ) |
16 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( 𝑀 ‘ 𝐵 ) ) |
17 |
12 13 15 9 10 11 16
|
reu2eqd |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |