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 |
1 2 3 4 5 6 7 8
|
lmif |
⊢ ( 𝜑 → 𝑀 : 𝑃 ⟶ 𝑃 ) |
10 |
9
|
ffnd |
⊢ ( 𝜑 → 𝑀 Fn 𝑃 ) |
11 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝐺 ∈ TarskiG ) |
12 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝐺 DimTarskiG≥ 2 ) |
13 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝐷 ∈ ran 𝐿 ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝑏 ∈ 𝑃 ) |
15 |
1 2 3 11 12 6 7 13 14
|
lmilmi |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 ) |
16 |
15
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑏 ∈ 𝑃 ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 ) |
17 |
|
nvocnv |
⊢ ( ( 𝑀 : 𝑃 ⟶ 𝑃 ∧ ∀ 𝑏 ∈ 𝑃 ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 ) → ◡ 𝑀 = 𝑀 ) |
18 |
9 16 17
|
syl2anc |
⊢ ( 𝜑 → ◡ 𝑀 = 𝑀 ) |
19 |
|
nvof1o |
⊢ ( ( 𝑀 Fn 𝑃 ∧ ◡ 𝑀 = 𝑀 ) → 𝑀 : 𝑃 –1-1-onto→ 𝑃 ) |
20 |
10 18 19
|
syl2anc |
⊢ ( 𝜑 → 𝑀 : 𝑃 –1-1-onto→ 𝑃 ) |