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 |
|
mirconn.m |
⊢ 𝑀 = ( 𝑆 ‘ 𝐴 ) |
8 |
|
mirconn.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
9 |
|
mirconn.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑃 ) |
10 |
|
mirconn.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑃 ) |
11 |
|
mirconn.1 |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ) |
12 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝐺 ∈ TarskiG ) |
13 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝑋 ∈ 𝑃 ) |
14 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝐴 ∈ 𝑃 ) |
15 |
1 2 3 4 5 6 8 7 10
|
mircl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ∈ 𝑃 ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → ( 𝑀 ‘ 𝑌 ) ∈ 𝑃 ) |
17 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝑌 ∈ 𝑃 ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) |
19 |
1 2 3 4 5 6 8 7 10
|
mirbtwn |
⊢ ( 𝜑 → 𝐴 ∈ ( ( 𝑀 ‘ 𝑌 ) 𝐼 𝑌 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝐴 ∈ ( ( 𝑀 ‘ 𝑌 ) 𝐼 𝑌 ) ) |
21 |
1 2 3 12 13 14 16 17 18 20
|
tgbtwnintr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐴 𝐼 𝑌 ) ) → 𝐴 ∈ ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
22 |
1 2 3 6 9 8
|
tgbtwntriv2 |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝑋 𝐼 𝐴 ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → 𝐴 ∈ ( 𝑋 𝐼 𝐴 ) ) |
24 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → 𝑌 = 𝐴 ) |
25 |
24
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝐴 ) ) |
26 |
1 2 3 4 5 6 8 7
|
mircinv |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = 𝐴 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → ( 𝑀 ‘ 𝐴 ) = 𝐴 ) |
28 |
25 27
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → ( 𝑀 ‘ 𝑌 ) = 𝐴 ) |
29 |
28
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) = ( 𝑋 𝐼 𝐴 ) ) |
30 |
23 29
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝐴 ) → 𝐴 ∈ ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
31 |
30
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 = 𝐴 ) → 𝐴 ∈ ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
32 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝐺 ∈ TarskiG ) |
33 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝑋 ∈ 𝑃 ) |
34 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝑌 ∈ 𝑃 ) |
35 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝐴 ∈ 𝑃 ) |
36 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → ( 𝑀 ‘ 𝑌 ) ∈ 𝑃 ) |
37 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝑌 ≠ 𝐴 ) |
38 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) |
39 |
1 2 3 32 35 34 33 38
|
tgbtwncom |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝑌 ∈ ( 𝑋 𝐼 𝐴 ) ) |
40 |
1 2 3 6 15 8 10 19
|
tgbtwncom |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝑌 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
41 |
40
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝐴 ∈ ( 𝑌 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
42 |
1 2 3 32 33 34 35 36 37 39 41
|
tgbtwnouttr2 |
⊢ ( ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) ∧ 𝑌 ≠ 𝐴 ) → 𝐴 ∈ ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
43 |
31 42
|
pm2.61dane |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝐴 𝐼 𝑋 ) ) → 𝐴 ∈ ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
44 |
21 43 11
|
mpjaodan |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝑋 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |