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 |
|
df-mir |
⊢ pInvG = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( ℩ 𝑧 ∈ ( Base ‘ 𝑔 ) ( ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) ) ) ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) ) |
10 |
9 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝑃 ) |
11 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( dist ‘ 𝑔 ) = ( dist ‘ 𝐺 ) ) |
12 |
11 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( dist ‘ 𝑔 ) = − ) |
13 |
12
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 − 𝑧 ) ) |
14 |
12
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) = ( 𝑥 − 𝑦 ) ) |
15 |
13 14
|
eqeq12d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) ↔ ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Itv ‘ 𝑔 ) = ( Itv ‘ 𝐺 ) ) |
17 |
16 3
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Itv ‘ 𝑔 ) = 𝐼 ) |
18 |
17
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) = ( 𝑧 𝐼 𝑦 ) ) |
19 |
18
|
eleq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ∈ ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) ↔ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) |
20 |
15 19
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) ) ↔ ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) |
21 |
10 20
|
riotaeqbidv |
⊢ ( 𝑔 = 𝐺 → ( ℩ 𝑧 ∈ ( Base ‘ 𝑔 ) ( ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) ) ) = ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) |
22 |
10 21
|
mpteq12dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( ℩ 𝑧 ∈ ( Base ‘ 𝑔 ) ( ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) ) ) ) = ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) |
23 |
10 22
|
mpteq12dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( ℩ 𝑧 ∈ ( Base ‘ 𝑔 ) ( ( 𝑥 ( dist ‘ 𝑔 ) 𝑧 ) = ( 𝑥 ( dist ‘ 𝑔 ) 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 ( Itv ‘ 𝑔 ) 𝑦 ) ) ) ) ) = ( 𝑥 ∈ 𝑃 ↦ ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) ) |
24 |
6
|
elexd |
⊢ ( 𝜑 → 𝐺 ∈ V ) |
25 |
1
|
fvexi |
⊢ 𝑃 ∈ V |
26 |
25
|
mptex |
⊢ ( 𝑥 ∈ 𝑃 ↦ ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) ∈ V |
27 |
26
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑃 ↦ ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) ∈ V ) |
28 |
8 23 24 27
|
fvmptd3 |
⊢ ( 𝜑 → ( pInvG ‘ 𝐺 ) = ( 𝑥 ∈ 𝑃 ↦ ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) ) |
29 |
5 28
|
syl5eq |
⊢ ( 𝜑 → 𝑆 = ( 𝑥 ∈ 𝑃 ↦ ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) ) |
30 |
|
simpll |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑧 ∈ 𝑃 ) → 𝑥 = 𝐴 ) |
31 |
30
|
oveq1d |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑧 ∈ 𝑃 ) → ( 𝑥 − 𝑧 ) = ( 𝐴 − 𝑧 ) ) |
32 |
30
|
oveq1d |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑧 ∈ 𝑃 ) → ( 𝑥 − 𝑦 ) = ( 𝐴 − 𝑦 ) ) |
33 |
31 32
|
eqeq12d |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑧 ∈ 𝑃 ) → ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ↔ ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ) ) |
34 |
30
|
eleq1d |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑧 ∈ 𝑃 ) → ( 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ↔ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) |
35 |
33 34
|
anbi12d |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑧 ∈ 𝑃 ) → ( ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ↔ ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) |
36 |
35
|
riotabidva |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃 ) → ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) = ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) |
37 |
36
|
mpteq2dva |
⊢ ( 𝑥 = 𝐴 → ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) = ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) |
38 |
37
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝑥 − 𝑧 ) = ( 𝑥 − 𝑦 ) ∧ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) = ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) |
39 |
25
|
mptex |
⊢ ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ∈ V |
40 |
39
|
a1i |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ∈ V ) |
41 |
29 38 7 40
|
fvmptd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) ) |