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 |
|
mirval.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
8 |
|
mirfv.m |
⊢ 𝑀 = ( 𝑆 ‘ 𝐴 ) |
9 |
|
mirinv.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
10 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐺 ∈ TarskiG ) |
11 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐵 ∈ 𝑃 ) |
12 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐴 ∈ 𝑃 ) |
13 |
1 2 3 4 5 10 12 8 11
|
mirbtwn |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐴 ∈ ( ( 𝑀 ‘ 𝐵 ) 𝐼 𝐵 ) ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → ( 𝑀 ‘ 𝐵 ) = 𝐵 ) |
15 |
14
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → ( ( 𝑀 ‘ 𝐵 ) 𝐼 𝐵 ) = ( 𝐵 𝐼 𝐵 ) ) |
16 |
13 15
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐴 ∈ ( 𝐵 𝐼 𝐵 ) ) |
17 |
1 2 3 10 11 12 16
|
axtgbtwnid |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐵 = 𝐴 ) |
18 |
17
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐴 = 𝐵 ) |
19 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐺 ∈ TarskiG ) |
20 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝑃 ) |
21 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝑃 ) |
22 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐵 ) ) |
23 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 ) |
24 |
1 2 3 19 21 21
|
tgbtwntriv1 |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ ( 𝐵 𝐼 𝐵 ) ) |
25 |
23 24
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ ( 𝐵 𝐼 𝐵 ) ) |
26 |
1 2 3 4 5 19 20 8 21 21 22 25
|
ismir |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 = ( 𝑀 ‘ 𝐵 ) ) |
27 |
26
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝑀 ‘ 𝐵 ) = 𝐵 ) |
28 |
18 27
|
impbida |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐵 ) = 𝐵 ↔ 𝐴 = 𝐵 ) ) |