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 |
|
mirfv.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
10 |
|
ismir.1 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
11 |
|
ismir.2 |
⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐴 − 𝐵 ) ) |
12 |
|
ismir.3 |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) |
13 |
1 2 3 4 5 6 7 8 9
|
mirfv |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) ) |
14 |
1 2 3 6 9 7
|
mirreu3 |
⊢ ( 𝜑 → ∃! 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐶 ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐴 − 𝐵 ) ) ) |
17 |
|
oveq1 |
⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐼 𝐵 ) = ( 𝐶 𝐼 𝐵 ) ) |
18 |
17
|
eleq2d |
⊢ ( 𝑧 = 𝐶 → ( 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ↔ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ) |
19 |
16 18
|
anbi12d |
⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ↔ ( ( 𝐴 − 𝐶 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ) ) |
20 |
19
|
riota2 |
⊢ ( ( 𝐶 ∈ 𝑃 ∧ ∃! 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) → ( ( ( 𝐴 − 𝐶 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ↔ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) = 𝐶 ) ) |
21 |
10 14 20
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐶 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ↔ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) = 𝐶 ) ) |
22 |
11 12 21
|
mpbi2and |
⊢ ( 𝜑 → ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) = 𝐶 ) |
23 |
13 22
|
eqtr2d |
⊢ ( 𝜑 → 𝐶 = ( 𝑀 ‘ 𝐵 ) ) |