Step |
Hyp |
Ref |
Expression |
1 |
|
hlpasch.p |
|- P = ( Base ` G ) |
2 |
|
hlpasch.i |
|- I = ( Itv ` G ) |
3 |
|
hlpasch.k |
|- K = ( hlG ` G ) |
4 |
|
hlpasch.g |
|- ( ph -> G e. TarskiG ) |
5 |
|
hlpasch.1 |
|- ( ph -> A e. P ) |
6 |
|
hlpasch.2 |
|- ( ph -> B e. P ) |
7 |
|
hlpasch.3 |
|- ( ph -> C e. P ) |
8 |
|
hlpasch.4 |
|- ( ph -> X e. P ) |
9 |
|
hlpasch.5 |
|- ( ph -> D e. P ) |
10 |
|
hlpasch.6 |
|- ( ph -> A =/= B ) |
11 |
|
hlpasch.7 |
|- ( ph -> C ( K ` B ) D ) |
12 |
|
hlpasch.8 |
|- ( ph -> A e. ( X I C ) ) |
13 |
|
eqid |
|- ( LineG ` G ) = ( LineG ` G ) |
14 |
4
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> G e. TarskiG ) |
15 |
9
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> D e. P ) |
16 |
8
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> X e. P ) |
17 |
7
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> C e. P ) |
18 |
6
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> B e. P ) |
19 |
5
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> A e. P ) |
20 |
|
eqid |
|- ( dist ` G ) = ( dist ` G ) |
21 |
|
simpr |
|- ( ( ph /\ C e. ( B I D ) ) -> C e. ( B I D ) ) |
22 |
1 20 2 14 18 17 15 21
|
tgbtwncom |
|- ( ( ph /\ C e. ( B I D ) ) -> C e. ( D I B ) ) |
23 |
12
|
adantr |
|- ( ( ph /\ C e. ( B I D ) ) -> A e. ( X I C ) ) |
24 |
1 2 13 14 15 16 17 18 19 22 23
|
outpasch |
|- ( ( ph /\ C e. ( B I D ) ) -> E. e e. P ( e e. ( D I X ) /\ A e. ( B I e ) ) ) |
25 |
|
simplr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> e e. P ) |
26 |
18
|
ad2antrr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> B e. P ) |
27 |
19
|
ad2antrr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> A e. P ) |
28 |
14
|
ad2antrr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> G e. TarskiG ) |
29 |
|
simprr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> A e. ( B I e ) ) |
30 |
1 20 2 28 26 27 25 29
|
tgbtwncom |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> A e. ( e I B ) ) |
31 |
28
|
adantr |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> G e. TarskiG ) |
32 |
26
|
adantr |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> B e. P ) |
33 |
27
|
adantr |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> A e. P ) |
34 |
|
simplrr |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> A e. ( B I e ) ) |
35 |
|
simpr |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> e = B ) |
36 |
35
|
oveq2d |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> ( B I e ) = ( B I B ) ) |
37 |
34 36
|
eleqtrd |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> A e. ( B I B ) ) |
38 |
1 20 2 31 32 33 37
|
axtgbtwnid |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> B = A ) |
39 |
38
|
eqcomd |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> A = B ) |
40 |
10
|
ad3antrrr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> A =/= B ) |
41 |
40
|
adantr |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> A =/= B ) |
42 |
41
|
neneqd |
|- ( ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) /\ e = B ) -> -. A = B ) |
43 |
39 42
|
pm2.65da |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> -. e = B ) |
44 |
43
|
neqned |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> e =/= B ) |
45 |
1 2 3 25 26 27 28 27 30 44 40
|
btwnhl2 |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> A ( K ` B ) e ) |
46 |
15
|
ad2antrr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> D e. P ) |
47 |
16
|
ad2antrr |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> X e. P ) |
48 |
|
simprl |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> e e. ( D I X ) ) |
49 |
1 20 2 28 46 25 47 48
|
tgbtwncom |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> e e. ( X I D ) ) |
50 |
45 49
|
jca |
|- ( ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) /\ ( e e. ( D I X ) /\ A e. ( B I e ) ) ) -> ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
51 |
50
|
ex |
|- ( ( ( ph /\ C e. ( B I D ) ) /\ e e. P ) -> ( ( e e. ( D I X ) /\ A e. ( B I e ) ) -> ( A ( K ` B ) e /\ e e. ( X I D ) ) ) ) |
52 |
51
|
reximdva |
|- ( ( ph /\ C e. ( B I D ) ) -> ( E. e e. P ( e e. ( D I X ) /\ A e. ( B I e ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) ) |
53 |
24 52
|
mpd |
|- ( ( ph /\ C e. ( B I D ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
54 |
9
|
ad2antrr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> D e. P ) |
55 |
54
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> D e. P ) |
56 |
|
simpr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) /\ e = D ) -> e = D ) |
57 |
56
|
breq2d |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) /\ e = D ) -> ( A ( K ` B ) e <-> A ( K ` B ) D ) ) |
58 |
56
|
eleq1d |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) /\ e = D ) -> ( e e. ( X I D ) <-> D e. ( X I D ) ) ) |
59 |
57 58
|
anbi12d |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) /\ e = D ) -> ( ( A ( K ` B ) e /\ e e. ( X I D ) ) <-> ( A ( K ` B ) D /\ D e. ( X I D ) ) ) ) |
60 |
5
|
ad2antrr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> A e. P ) |
61 |
60
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> A e. P ) |
62 |
6
|
ad2antrr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> B e. P ) |
63 |
62
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> B e. P ) |
64 |
4
|
ad2antrr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> G e. TarskiG ) |
65 |
64
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> G e. TarskiG ) |
66 |
1 2 3 7 9 6 4 11
|
hlcomd |
|- ( ph -> D ( K ` B ) C ) |
67 |
66
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> D ( K ` B ) C ) |
68 |
7
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> C e. P ) |
69 |
68
|
ad2antrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> C e. P ) |
70 |
12
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> A e. ( X I C ) ) |
71 |
70
|
ad2antrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> A e. ( X I C ) ) |
72 |
|
simpr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> X = B ) |
73 |
72
|
oveq1d |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> ( X I C ) = ( B I C ) ) |
74 |
71 73
|
eleqtrd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> A e. ( B I C ) ) |
75 |
1 2 3 7 9 6 4
|
ishlg |
|- ( ph -> ( C ( K ` B ) D <-> ( C =/= B /\ D =/= B /\ ( C e. ( B I D ) \/ D e. ( B I C ) ) ) ) ) |
76 |
11 75
|
mpbid |
|- ( ph -> ( C =/= B /\ D =/= B /\ ( C e. ( B I D ) \/ D e. ( B I C ) ) ) ) |
77 |
76
|
simp1d |
|- ( ph -> C =/= B ) |
78 |
77
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> C =/= B ) |
79 |
10
|
ad2antrr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> A =/= B ) |
80 |
79
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> A =/= B ) |
81 |
1 2 3 55 69 63 65 61 74 78 80
|
hlbtwn |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> ( D ( K ` B ) C <-> D ( K ` B ) A ) ) |
82 |
67 81
|
mpbid |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> D ( K ` B ) A ) |
83 |
1 2 3 55 61 63 65 82
|
hlcomd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> A ( K ` B ) D ) |
84 |
8
|
ad2antrr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> X e. P ) |
85 |
84
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> X e. P ) |
86 |
1 20 2 65 85 55
|
tgbtwntriv2 |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> D e. ( X I D ) ) |
87 |
83 86
|
jca |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> ( A ( K ` B ) D /\ D e. ( X I D ) ) ) |
88 |
55 59 87
|
rspcedvd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X = B ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
89 |
84
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ A ( K ` B ) X ) -> X e. P ) |
90 |
|
simpr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ e = X ) -> e = X ) |
91 |
90
|
breq2d |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ e = X ) -> ( A ( K ` B ) e <-> A ( K ` B ) X ) ) |
92 |
90
|
eleq1d |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ e = X ) -> ( e e. ( X I D ) <-> X e. ( X I D ) ) ) |
93 |
91 92
|
anbi12d |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ e = X ) -> ( ( A ( K ` B ) e /\ e e. ( X I D ) ) <-> ( A ( K ` B ) X /\ X e. ( X I D ) ) ) ) |
94 |
93
|
ad4ant14 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ A ( K ` B ) X ) /\ e = X ) -> ( ( A ( K ` B ) e /\ e e. ( X I D ) ) <-> ( A ( K ` B ) X /\ X e. ( X I D ) ) ) ) |
95 |
|
simpr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ A ( K ` B ) X ) -> A ( K ` B ) X ) |
96 |
1 20 2 64 84 54
|
tgbtwntriv1 |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> X e. ( X I D ) ) |
97 |
96
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ A ( K ` B ) X ) -> X e. ( X I D ) ) |
98 |
95 97
|
jca |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ A ( K ` B ) X ) -> ( A ( K ` B ) X /\ X e. ( X I D ) ) ) |
99 |
89 94 98
|
rspcedvd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ A ( K ` B ) X ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
100 |
54
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> D e. P ) |
101 |
|
simpr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) /\ e = D ) -> e = D ) |
102 |
101
|
breq2d |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) /\ e = D ) -> ( A ( K ` B ) e <-> A ( K ` B ) D ) ) |
103 |
101
|
eleq1d |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) /\ e = D ) -> ( e e. ( X I D ) <-> D e. ( X I D ) ) ) |
104 |
102 103
|
anbi12d |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) /\ e = D ) -> ( ( A ( K ` B ) e /\ e e. ( X I D ) ) <-> ( A ( K ` B ) D /\ D e. ( X I D ) ) ) ) |
105 |
79
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> A =/= B ) |
106 |
1 2 3 7 9 6 4 11
|
hlne2 |
|- ( ph -> D =/= B ) |
107 |
106
|
ad4antr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> D =/= B ) |
108 |
64
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> G e. TarskiG ) |
109 |
62
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> B e. P ) |
110 |
60
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> A e. P ) |
111 |
68
|
ad2antrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> C e. P ) |
112 |
111
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> C e. P ) |
113 |
84
|
ad2antrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> X e. P ) |
114 |
|
simpr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> B e. ( X I A ) ) |
115 |
70
|
ad2antrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> A e. ( X I C ) ) |
116 |
115
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> A e. ( X I C ) ) |
117 |
1 20 2 108 113 109 110 112 114 116
|
tgbtwnexch3 |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> A e. ( B I C ) ) |
118 |
|
simp-4r |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> D e. ( B I C ) ) |
119 |
1 2 108 109 110 100 112 117 118
|
tgbtwnconn3 |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> ( A e. ( B I D ) \/ D e. ( B I A ) ) ) |
120 |
1 2 3 5 9 6 4
|
ishlg |
|- ( ph -> ( A ( K ` B ) D <-> ( A =/= B /\ D =/= B /\ ( A e. ( B I D ) \/ D e. ( B I A ) ) ) ) ) |
121 |
120
|
ad4antr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> ( A ( K ` B ) D <-> ( A =/= B /\ D =/= B /\ ( A e. ( B I D ) \/ D e. ( B I A ) ) ) ) ) |
122 |
105 107 119 121
|
mpbir3and |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> A ( K ` B ) D ) |
123 |
1 20 2 108 113 100
|
tgbtwntriv2 |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> D e. ( X I D ) ) |
124 |
122 123
|
jca |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> ( A ( K ` B ) D /\ D e. ( X I D ) ) ) |
125 |
100 104 124
|
rspcedvd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ B e. ( X I A ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
126 |
8
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> X e. P ) |
127 |
6
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> B e. P ) |
128 |
5
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> A e. P ) |
129 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> G e. TarskiG ) |
130 |
|
simpr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> X =/= B ) |
131 |
130
|
neneqd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> -. X = B ) |
132 |
64
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> G e. TarskiG ) |
133 |
132
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> G e. TarskiG ) |
134 |
126
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> X e. P ) |
135 |
128
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> A e. P ) |
136 |
115
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> A e. ( X I C ) ) |
137 |
|
simpr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> X = C ) |
138 |
137
|
oveq2d |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> ( X I X ) = ( X I C ) ) |
139 |
136 138
|
eleqtrrd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> A e. ( X I X ) ) |
140 |
1 20 2 133 134 135 139
|
axtgbtwnid |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> X = A ) |
141 |
140
|
olcd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X = C ) -> ( B e. ( X ( LineG ` G ) A ) \/ X = A ) ) |
142 |
132
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> G e. TarskiG ) |
143 |
127
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> B e. P ) |
144 |
111
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> C e. P ) |
145 |
126
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> X e. P ) |
146 |
128
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> A e. P ) |
147 |
|
simpr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> X =/= C ) |
148 |
147
|
necomd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> C =/= X ) |
149 |
148
|
neneqd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> -. C = X ) |
150 |
54
|
adantr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> D e. P ) |
151 |
106
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> D =/= B ) |
152 |
|
simplr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> X e. ( B ( LineG ` G ) D ) ) |
153 |
1 2 13 132 150 127 126 151 152
|
lncom |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> X e. ( D ( LineG ` G ) B ) ) |
154 |
77
|
necomd |
|- ( ph -> B =/= C ) |
155 |
154
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> B =/= C ) |
156 |
66
|
ad3antrrr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> D ( K ` B ) C ) |
157 |
1 2 3 150 111 127 132 13 156
|
hlln |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> D e. ( C ( LineG ` G ) B ) ) |
158 |
1 2 13 132 127 111 150 155 157
|
lncom |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> D e. ( B ( LineG ` G ) C ) ) |
159 |
158
|
orcd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( D e. ( B ( LineG ` G ) C ) \/ B = C ) ) |
160 |
1 2 13 132 126 150 127 111 153 159
|
coltr |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( X e. ( B ( LineG ` G ) C ) \/ B = C ) ) |
161 |
1 13 2 132 127 111 126 160
|
colrot1 |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( B e. ( C ( LineG ` G ) X ) \/ C = X ) ) |
162 |
161
|
orcomd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( C = X \/ B e. ( C ( LineG ` G ) X ) ) ) |
163 |
162
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> ( C = X \/ B e. ( C ( LineG ` G ) X ) ) ) |
164 |
163
|
ord |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> ( -. C = X -> B e. ( C ( LineG ` G ) X ) ) ) |
165 |
149 164
|
mpd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> B e. ( C ( LineG ` G ) X ) ) |
166 |
1 13 2 132 126 128 111 115
|
btwncolg3 |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( C e. ( X ( LineG ` G ) A ) \/ X = A ) ) |
167 |
166
|
adantr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> ( C e. ( X ( LineG ` G ) A ) \/ X = A ) ) |
168 |
1 2 13 142 143 144 145 146 165 167
|
coltr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) /\ X =/= C ) -> ( B e. ( X ( LineG ` G ) A ) \/ X = A ) ) |
169 |
141 168
|
pm2.61dane |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( B e. ( X ( LineG ` G ) A ) \/ X = A ) ) |
170 |
1 13 2 132 126 128 127 169
|
colrot2 |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( A e. ( B ( LineG ` G ) X ) \/ B = X ) ) |
171 |
1 13 2 132 127 126 128 170
|
colcom |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( A e. ( X ( LineG ` G ) B ) \/ X = B ) ) |
172 |
171
|
orcomd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( X = B \/ A e. ( X ( LineG ` G ) B ) ) ) |
173 |
172
|
ord |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( -. X = B -> A e. ( X ( LineG ` G ) B ) ) ) |
174 |
131 173
|
mpd |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> A e. ( X ( LineG ` G ) B ) ) |
175 |
1 2 3 126 127 128 129 128 13 174
|
lnhl |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> ( A ( K ` B ) X \/ B e. ( X I A ) ) ) |
176 |
99 125 175
|
mpjaodan |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) /\ X =/= B ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
177 |
88 176
|
pm2.61dane |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ X e. ( B ( LineG ` G ) D ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
178 |
4
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> G e. TarskiG ) |
179 |
8
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> X e. P ) |
180 |
6
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> B e. P ) |
181 |
5
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> A e. P ) |
182 |
9
|
adantr |
|- ( ( ph /\ D e. ( B I C ) ) -> D e. P ) |
183 |
|
simpr |
|- ( ( ph /\ D e. ( B I C ) ) -> D e. ( B I C ) ) |
184 |
1 20 2 178 179 180 68 181 182 70 183
|
axtgpasch |
|- ( ( ph /\ D e. ( B I C ) ) -> E. e e. P ( e e. ( A I B ) /\ e e. ( D I X ) ) ) |
185 |
184
|
adantr |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) -> E. e e. P ( e e. ( A I B ) /\ e e. ( D I X ) ) ) |
186 |
|
simplr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> e e. P ) |
187 |
181
|
ad3antrrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> A e. P ) |
188 |
180
|
ad3antrrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> B e. P ) |
189 |
178
|
ad3antrrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> G e. TarskiG ) |
190 |
|
simprl |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> e e. ( A I B ) ) |
191 |
1 20 2 189 187 186 188 190
|
tgbtwncom |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> e e. ( B I A ) ) |
192 |
10
|
necomd |
|- ( ph -> B =/= A ) |
193 |
192
|
ad4antr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> B =/= A ) |
194 |
189
|
adantr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> G e. TarskiG ) |
195 |
9
|
ad5antr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> D e. P ) |
196 |
8
|
ad5antr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> X e. P ) |
197 |
188
|
adantr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> B e. P ) |
198 |
|
simp-4r |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> -. X e. ( B ( LineG ` G ) D ) ) |
199 |
106
|
necomd |
|- ( ph -> B =/= D ) |
200 |
199
|
ad5antr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> B =/= D ) |
201 |
200
|
neneqd |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> -. B = D ) |
202 |
|
ioran |
|- ( -. ( X e. ( B ( LineG ` G ) D ) \/ B = D ) <-> ( -. X e. ( B ( LineG ` G ) D ) /\ -. B = D ) ) |
203 |
198 201 202
|
sylanbrc |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> -. ( X e. ( B ( LineG ` G ) D ) \/ B = D ) ) |
204 |
1 13 2 194 197 195 196 203
|
ncolrot2 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> -. ( D e. ( X ( LineG ` G ) B ) \/ X = B ) ) |
205 |
|
simpr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> e = B ) |
206 |
186
|
adantr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> e e. P ) |
207 |
1 2 13 194 195 196 197 204
|
ncolne1 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> D =/= X ) |
208 |
|
simplrr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> e e. ( D I X ) ) |
209 |
1 2 13 194 195 196 206 207 208
|
btwnlng1 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> e e. ( D ( LineG ` G ) X ) ) |
210 |
205 209
|
eqeltrrd |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> B e. ( D ( LineG ` G ) X ) ) |
211 |
1 2 13 194 195 196 207
|
tglinerflx1 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> D e. ( D ( LineG ` G ) X ) ) |
212 |
106
|
ad5antr |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> D =/= B ) |
213 |
212
|
necomd |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> B =/= D ) |
214 |
1 2 13 194 197 195 213
|
tglinerflx1 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> B e. ( B ( LineG ` G ) D ) ) |
215 |
1 2 13 194 197 195 213
|
tglinerflx2 |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> D e. ( B ( LineG ` G ) D ) ) |
216 |
1 2 13 194 195 196 197 195 204 210 211 214 215
|
tglineinteq |
|- ( ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) /\ e = B ) -> B = D ) |
217 |
216 201
|
pm2.65da |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> -. e = B ) |
218 |
217
|
neqned |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> e =/= B ) |
219 |
1 2 3 188 187 186 189 187 191 193 218
|
btwnhl1 |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> e ( K ` B ) A ) |
220 |
1 2 3 186 187 188 189 219
|
hlcomd |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> A ( K ` B ) e ) |
221 |
178
|
ad3antrrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ e e. ( D I X ) ) -> G e. TarskiG ) |
222 |
182
|
ad3antrrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ e e. ( D I X ) ) -> D e. P ) |
223 |
|
simplr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ e e. ( D I X ) ) -> e e. P ) |
224 |
179
|
ad3antrrr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ e e. ( D I X ) ) -> X e. P ) |
225 |
|
simpr |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ e e. ( D I X ) ) -> e e. ( D I X ) ) |
226 |
1 20 2 221 222 223 224 225
|
tgbtwncom |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ e e. ( D I X ) ) -> e e. ( X I D ) ) |
227 |
226
|
adantrl |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> e e. ( X I D ) ) |
228 |
220 227
|
jca |
|- ( ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) /\ ( e e. ( A I B ) /\ e e. ( D I X ) ) ) -> ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
229 |
228
|
ex |
|- ( ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) /\ e e. P ) -> ( ( e e. ( A I B ) /\ e e. ( D I X ) ) -> ( A ( K ` B ) e /\ e e. ( X I D ) ) ) ) |
230 |
229
|
reximdva |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) -> ( E. e e. P ( e e. ( A I B ) /\ e e. ( D I X ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) ) |
231 |
185 230
|
mpd |
|- ( ( ( ph /\ D e. ( B I C ) ) /\ -. X e. ( B ( LineG ` G ) D ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
232 |
177 231
|
pm2.61dan |
|- ( ( ph /\ D e. ( B I C ) ) -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |
233 |
76
|
simp3d |
|- ( ph -> ( C e. ( B I D ) \/ D e. ( B I C ) ) ) |
234 |
53 232 233
|
mpjaodan |
|- ( ph -> E. e e. P ( A ( K ` B ) e /\ e e. ( X I D ) ) ) |