Step |
Hyp |
Ref |
Expression |
1 |
|
israg.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
israg.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
israg.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
israg.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
5 |
|
israg.s |
⊢ 𝑆 = ( pInvG ‘ 𝐺 ) |
6 |
|
israg.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
7 |
|
israg.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
8 |
|
israg.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
9 |
|
israg.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
10 |
|
ragcgr.c |
⊢ ∼ = ( cgrG ‘ 𝐺 ) |
11 |
|
ragcgr.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
12 |
|
ragcgr.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) |
13 |
|
ragcgr.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) |
14 |
|
ragcgr.1 |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
15 |
|
ragcgr.2 |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ∼ 〈“ 𝐷 𝐸 𝐹 ”〉 ) |
16 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐷 = 𝐷 ) |
17 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐺 ∈ TarskiG ) |
18 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐵 ∈ 𝑃 ) |
19 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐶 ∈ 𝑃 ) |
20 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐸 ∈ 𝑃 ) |
21 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐹 ∈ 𝑃 ) |
22 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐴 ∈ 𝑃 ) |
23 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐷 ∈ 𝑃 ) |
24 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 〈“ 𝐴 𝐵 𝐶 ”〉 ∼ 〈“ 𝐷 𝐸 𝐹 ”〉 ) |
25 |
1 2 3 10 17 22 18 19 23 20 21 24
|
cgr3simp2 |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ) |
26 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐵 = 𝐶 ) |
27 |
1 2 3 17 18 19 20 21 25 26
|
tgcgreq |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐸 = 𝐹 ) |
28 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐹 = 𝐹 ) |
29 |
16 27 28
|
s3eqd |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 〈“ 𝐷 𝐸 𝐹 ”〉 = 〈“ 𝐷 𝐹 𝐹 ”〉 ) |
30 |
1 2 3 4 5 17 23 21 20
|
ragtrivb |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 〈“ 𝐷 𝐹 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
31 |
29 30
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 〈“ 𝐷 𝐸 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
32 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
33 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐺 ∈ TarskiG ) |
34 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐴 ∈ 𝑃 ) |
35 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ 𝑃 ) |
36 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐶 ∈ 𝑃 ) |
37 |
1 2 3 4 5 33 34 35 36
|
israg |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) ) |
38 |
32 37
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) |
39 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐹 ∈ 𝑃 ) |
40 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐷 ∈ 𝑃 ) |
41 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐸 ∈ 𝑃 ) |
42 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 〈“ 𝐴 𝐵 𝐶 ”〉 ∼ 〈“ 𝐷 𝐸 𝐹 ”〉 ) |
43 |
1 2 3 10 33 34 35 36 40 41 39 42
|
cgr3simp3 |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) |
44 |
1 2 3 33 36 34 39 40 43
|
tgcgrcomlr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
45 |
|
eqid |
⊢ ( 𝑆 ‘ 𝐵 ) = ( 𝑆 ‘ 𝐵 ) |
46 |
1 2 3 4 5 33 35 45 36
|
mircl |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ∈ 𝑃 ) |
47 |
|
eqid |
⊢ ( 𝑆 ‘ 𝐸 ) = ( 𝑆 ‘ 𝐸 ) |
48 |
1 2 3 4 5 33 41 47 39
|
mircl |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ∈ 𝑃 ) |
49 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ≠ 𝐶 ) |
50 |
49
|
necomd |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐶 ≠ 𝐵 ) |
51 |
1 2 3 4 5 33 35 45 36
|
mirbtwn |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ ( ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) 𝐼 𝐶 ) ) |
52 |
1 2 3 33 46 35 36 51
|
tgbtwncom |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ ( 𝐶 𝐼 ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) |
53 |
1 2 3 4 5 33 41 47 39
|
mirbtwn |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐸 ∈ ( ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) 𝐼 𝐹 ) ) |
54 |
1 2 3 33 48 41 39 53
|
tgbtwncom |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐸 ∈ ( 𝐹 𝐼 ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ) ) |
55 |
1 2 3 10 33 34 35 36 40 41 39 42
|
cgr3simp2 |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ) |
56 |
1 2 3 33 35 36 41 39 55
|
tgcgrcomlr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐶 − 𝐵 ) = ( 𝐹 − 𝐸 ) ) |
57 |
1 2 3 4 5 33 35 45 36
|
mircgr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐵 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = ( 𝐵 − 𝐶 ) ) |
58 |
1 2 3 4 5 33 41 47 39
|
mircgr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐸 − ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ) = ( 𝐸 − 𝐹 ) ) |
59 |
55 57 58
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐵 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = ( 𝐸 − ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ) ) |
60 |
1 2 3 10 33 34 35 36 40 41 39 42
|
cgr3simp1 |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ) |
61 |
1 2 3 33 34 35 40 41 60
|
tgcgrcomlr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐵 − 𝐴 ) = ( 𝐸 − 𝐷 ) ) |
62 |
1 2 3 33 36 35 46 39 41 48 34 40 50 52 54 56 59 43 61
|
axtg5seg |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) − 𝐴 ) = ( ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) − 𝐷 ) ) |
63 |
1 2 3 33 46 34 48 40 62
|
tgcgrcomlr |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = ( 𝐷 − ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ) ) |
64 |
38 44 63
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 𝐷 − 𝐹 ) = ( 𝐷 − ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ) ) |
65 |
1 2 3 4 5 33 40 41 39
|
israg |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → ( 〈“ 𝐷 𝐸 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐷 − 𝐹 ) = ( 𝐷 − ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐹 ) ) ) ) |
66 |
64 65
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 〈“ 𝐷 𝐸 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
67 |
31 66
|
pm2.61dane |
⊢ ( 𝜑 → 〈“ 𝐷 𝐸 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |