Step | Hyp | Ref
| Expression |
1 | | eqidd 2731 |
. . . 4
β’ ((π β§ π΅ = πΆ) β π· = π·) |
2 | | israg.p |
. . . . 5
β’ π = (BaseβπΊ) |
3 | | israg.d |
. . . . 5
β’ β =
(distβπΊ) |
4 | | israg.i |
. . . . 5
β’ πΌ = (ItvβπΊ) |
5 | | israg.g |
. . . . . 6
β’ (π β πΊ β TarskiG) |
6 | 5 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β πΊ β TarskiG) |
7 | | israg.b |
. . . . . 6
β’ (π β π΅ β π) |
8 | 7 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β π΅ β π) |
9 | | israg.c |
. . . . . 6
β’ (π β πΆ β π) |
10 | 9 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β πΆ β π) |
11 | | ragcgr.e |
. . . . . 6
β’ (π β πΈ β π) |
12 | 11 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β πΈ β π) |
13 | | ragcgr.f |
. . . . . 6
β’ (π β πΉ β π) |
14 | 13 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β πΉ β π) |
15 | | ragcgr.c |
. . . . . 6
β’ βΌ =
(cgrGβπΊ) |
16 | | israg.a |
. . . . . . 7
β’ (π β π΄ β π) |
17 | 16 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ = πΆ) β π΄ β π) |
18 | | ragcgr.d |
. . . . . . 7
β’ (π β π· β π) |
19 | 18 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ = πΆ) β π· β π) |
20 | | ragcgr.2 |
. . . . . . 7
β’ (π β β¨βπ΄π΅πΆββ© βΌ β¨βπ·πΈπΉββ©) |
21 | 20 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ = πΆ) β β¨βπ΄π΅πΆββ© βΌ β¨βπ·πΈπΉββ©) |
22 | 2, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21 | cgr3simp2 28039 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β (π΅ β πΆ) = (πΈ β πΉ)) |
23 | | simpr 483 |
. . . . 5
β’ ((π β§ π΅ = πΆ) β π΅ = πΆ) |
24 | 2, 3, 4, 6, 8, 10,
12, 14, 22, 23 | tgcgreq 28000 |
. . . 4
β’ ((π β§ π΅ = πΆ) β πΈ = πΉ) |
25 | | eqidd 2731 |
. . . 4
β’ ((π β§ π΅ = πΆ) β πΉ = πΉ) |
26 | 1, 24, 25 | s3eqd 14819 |
. . 3
β’ ((π β§ π΅ = πΆ) β β¨βπ·πΈπΉββ© = β¨βπ·πΉπΉββ©) |
27 | | israg.l |
. . . 4
β’ πΏ = (LineGβπΊ) |
28 | | israg.s |
. . . 4
β’ π = (pInvGβπΊ) |
29 | 2, 3, 4, 27, 28, 6, 19, 14, 12 | ragtrivb 28220 |
. . 3
β’ ((π β§ π΅ = πΆ) β β¨βπ·πΉπΉββ© β (βGβπΊ)) |
30 | 26, 29 | eqeltrd 2831 |
. 2
β’ ((π β§ π΅ = πΆ) β β¨βπ·πΈπΉββ© β (βGβπΊ)) |
31 | | ragcgr.1 |
. . . . . 6
β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) |
32 | 31 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β β¨βπ΄π΅πΆββ© β (βGβπΊ)) |
33 | 5 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β πΊ β TarskiG) |
34 | 16 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β π΄ β π) |
35 | 7 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β π΅ β π) |
36 | 9 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β πΆ β π) |
37 | 2, 3, 4, 27, 28, 33, 34, 35, 36 | israg 28215 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β (β¨βπ΄π΅πΆββ© β (βGβπΊ) β (π΄ β πΆ) = (π΄ β ((πβπ΅)βπΆ)))) |
38 | 32, 37 | mpbid 231 |
. . . 4
β’ ((π β§ π΅ β πΆ) β (π΄ β πΆ) = (π΄ β ((πβπ΅)βπΆ))) |
39 | 13 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β πΉ β π) |
40 | 18 | adantr 479 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β π· β π) |
41 | 11 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β πΈ β π) |
42 | 20 | adantr 479 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β β¨βπ΄π΅πΆββ© βΌ β¨βπ·πΈπΉββ©) |
43 | 2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42 | cgr3simp3 28040 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β (πΆ β π΄) = (πΉ β π·)) |
44 | 2, 3, 4, 33, 36, 34, 39, 40, 43 | tgcgrcomlr 27998 |
. . . 4
β’ ((π β§ π΅ β πΆ) β (π΄ β πΆ) = (π· β πΉ)) |
45 | | eqid 2730 |
. . . . . 6
β’ (πβπ΅) = (πβπ΅) |
46 | 2, 3, 4, 27, 28, 33, 35, 45, 36 | mircl 28179 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β ((πβπ΅)βπΆ) β π) |
47 | | eqid 2730 |
. . . . . 6
β’ (πβπΈ) = (πβπΈ) |
48 | 2, 3, 4, 27, 28, 33, 41, 47, 39 | mircl 28179 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β ((πβπΈ)βπΉ) β π) |
49 | | simpr 483 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β π΅ β πΆ) |
50 | 49 | necomd 2994 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β πΆ β π΅) |
51 | 2, 3, 4, 27, 28, 33, 35, 45, 36 | mirbtwn 28176 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β π΅ β (((πβπ΅)βπΆ)πΌπΆ)) |
52 | 2, 3, 4, 33, 46, 35, 36, 51 | tgbtwncom 28006 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β π΅ β (πΆπΌ((πβπ΅)βπΆ))) |
53 | 2, 3, 4, 27, 28, 33, 41, 47, 39 | mirbtwn 28176 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β πΈ β (((πβπΈ)βπΉ)πΌπΉ)) |
54 | 2, 3, 4, 33, 48, 41, 39, 53 | tgbtwncom 28006 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β πΈ β (πΉπΌ((πβπΈ)βπΉ))) |
55 | 2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42 | cgr3simp2 28039 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β (π΅ β πΆ) = (πΈ β πΉ)) |
56 | 2, 3, 4, 33, 35, 36, 41, 39, 55 | tgcgrcomlr 27998 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β (πΆ β π΅) = (πΉ β πΈ)) |
57 | 2, 3, 4, 27, 28, 33, 35, 45, 36 | mircgr 28175 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β (π΅ β ((πβπ΅)βπΆ)) = (π΅ β πΆ)) |
58 | 2, 3, 4, 27, 28, 33, 41, 47, 39 | mircgr 28175 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β (πΈ β ((πβπΈ)βπΉ)) = (πΈ β πΉ)) |
59 | 55, 57, 58 | 3eqtr4d 2780 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β (π΅ β ((πβπ΅)βπΆ)) = (πΈ β ((πβπΈ)βπΉ))) |
60 | 2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42 | cgr3simp1 28038 |
. . . . . . 7
β’ ((π β§ π΅ β πΆ) β (π΄ β π΅) = (π· β πΈ)) |
61 | 2, 3, 4, 33, 34, 35, 40, 41, 60 | tgcgrcomlr 27998 |
. . . . . 6
β’ ((π β§ π΅ β πΆ) β (π΅ β π΄) = (πΈ β π·)) |
62 | 2, 3, 4, 33, 36, 35, 46, 39, 41, 48, 34, 40, 50, 52, 54, 56, 59, 43, 61 | axtg5seg 27983 |
. . . . 5
β’ ((π β§ π΅ β πΆ) β (((πβπ΅)βπΆ) β π΄) = (((πβπΈ)βπΉ) β π·)) |
63 | 2, 3, 4, 33, 46, 34, 48, 40, 62 | tgcgrcomlr 27998 |
. . . 4
β’ ((π β§ π΅ β πΆ) β (π΄ β ((πβπ΅)βπΆ)) = (π· β ((πβπΈ)βπΉ))) |
64 | 38, 44, 63 | 3eqtr3d 2778 |
. . 3
β’ ((π β§ π΅ β πΆ) β (π· β πΉ) = (π· β ((πβπΈ)βπΉ))) |
65 | 2, 3, 4, 27, 28, 33, 40, 41, 39 | israg 28215 |
. . 3
β’ ((π β§ π΅ β πΆ) β (β¨βπ·πΈπΉββ© β (βGβπΊ) β (π· β πΉ) = (π· β ((πβπΈ)βπΉ)))) |
66 | 64, 65 | mpbird 256 |
. 2
β’ ((π β§ π΅ β πΆ) β β¨βπ·πΈπΉββ© β (βGβπΊ)) |
67 | 30, 66 | pm2.61dane 3027 |
1
β’ (π β β¨βπ·πΈπΉββ© β (βGβπΊ)) |