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 |
1 2 3 4 5 6 7 8 9
|
lmicl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ 𝑃 ) |
11 |
1 2 3 4 5 6 7 8 9
|
lmilmi |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝐴 ) ) = 𝐴 ) |
12 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → 𝐺 ∈ TarskiG ) |
13 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → 𝐺 DimTarskiG≥ 2 ) |
14 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → 𝐷 ∈ ran 𝐿 ) |
15 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → 𝑏 ∈ 𝑃 ) |
16 |
1 2 3 12 13 6 7 14 15
|
lmilmi |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 ) |
17 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → ( 𝑀 ‘ 𝑏 ) = 𝐴 ) |
18 |
17
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = ( 𝑀 ‘ 𝐴 ) ) |
19 |
16 18
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) ∧ ( 𝑀 ‘ 𝑏 ) = 𝐴 ) → 𝑏 = ( 𝑀 ‘ 𝐴 ) ) |
20 |
19
|
ex |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → ( ( 𝑀 ‘ 𝑏 ) = 𝐴 → 𝑏 = ( 𝑀 ‘ 𝐴 ) ) ) |
21 |
20
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑏 ) = 𝐴 → 𝑏 = ( 𝑀 ‘ 𝐴 ) ) ) |
22 |
|
fveqeq2 |
⊢ ( 𝑏 = ( 𝑀 ‘ 𝐴 ) → ( ( 𝑀 ‘ 𝑏 ) = 𝐴 ↔ ( 𝑀 ‘ ( 𝑀 ‘ 𝐴 ) ) = 𝐴 ) ) |
23 |
22
|
eqreu |
⊢ ( ( ( 𝑀 ‘ 𝐴 ) ∈ 𝑃 ∧ ( 𝑀 ‘ ( 𝑀 ‘ 𝐴 ) ) = 𝐴 ∧ ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑏 ) = 𝐴 → 𝑏 = ( 𝑀 ‘ 𝐴 ) ) ) → ∃! 𝑏 ∈ 𝑃 ( 𝑀 ‘ 𝑏 ) = 𝐴 ) |
24 |
10 11 21 23
|
syl3anc |
⊢ ( 𝜑 → ∃! 𝑏 ∈ 𝑃 ( 𝑀 ‘ 𝑏 ) = 𝐴 ) |