Theorem hypcgrlem2 25692

Description: Lemma for hypcgr 25693, case where triangles share one vertex B. (Contributed by Thierry Arnoux, 16-Dec-2019.)

Hypotheses

Ref	Expression
hypcgr.p	|- P = ( Base ` G )
hypcgr.m	|- .- = ( dist ` G )
hypcgr.i	|- I = (Itv ` G )
hypcgr.g	|- ( ph -> G e. TarskiG )
hypcgr.h	|- ( ph -> GDimTarskiG≥ 2 )
hypcgr.a	|- ( ph -> A e. P )
hypcgr.b	|- ( ph -> B e. P )
hypcgr.c	|- ( ph -> C e. P )
hypcgr.d	|- ( ph -> D e. P )
hypcgr.e	|- ( ph -> E e. P )
hypcgr.f	|- ( ph -> F e. P )
hypcgr.1	|- ( ph -> <" A B C "> e. (∟G ` G ) )
hypcgr.2	|- ( ph -> <" D E F "> e. (∟G ` G ) )
hypcgr.3	|- ( ph -> ( A .- B ) = ( D .- E ) )
hypcgr.4	|- ( ph -> ( B .- C ) = ( E .- F ) )
hypcgrlem2.b	|- ( ph -> B = E )
hypcgrlem2.s	|- S = ( (lInvG ` G ) ` ( ( C (midG ` G ) F ) (LineG ` G ) B ) )

Assertion

Ref	Expression
hypcgrlem2	|- ( ph -> ( A .- C ) = ( D .- F ) )

Proof of Theorem hypcgrlem2

Step	Hyp	Ref	Expression
1		hypcgr.p	. . . 4 |- P = ( Base ` G )
2		hypcgr.m	. . . 4 |- .- = ( dist ` G )
3		hypcgr.i	. . . 4 |- I = (Itv ` G )
4		hypcgr.g	. . . . 5 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> G e. TarskiG )
6		hypcgr.h	. . . . 5 |- ( ph -> GDimTarskiG≥ 2 )
7	6	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> GDimTarskiG≥ 2 )
8		hypcgr.a	. . . . 5 |- ( ph -> A e. P )
9	8	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> A e. P )
10		hypcgr.b	. . . . 5 |- ( ph -> B e. P )
11	10	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> B e. P )
12		hypcgr.c	. . . . 5 |- ( ph -> C e. P )
13	12	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> C e. P )
14		eqid 2622	. . . . 5 |- (LineG ` G ) = (LineG ` G )
15		eqid 2622	. . . . 5 |- (pInvG ` G ) = (pInvG ` G )
16		eqid 2622	. . . . 5 |- ( (pInvG ` G ) ` B ) = ( (pInvG ` G ) ` B )
17		hypcgr.d	. . . . . 6 |- ( ph -> D e. P )
18	17	adantr 481	. . . . 5 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> D e. P )
19	1, 2, 3, 14, 15, 5, 11, 16, 18	mircl 25556	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( (pInvG ` G ) ` B ) ` D ) e. P )
20		hypcgr.e	. . . . 5 |- ( ph -> E e. P )
21	20	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> E e. P )
22		hypcgr.1	. . . . 5 |- ( ph -> <" A B C "> e. (∟G ` G ) )
23	22	adantr 481	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> <" A B C "> e. (∟G ` G ) )
24		eqidd 2623	. . . . . 6 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( (pInvG ` G ) ` B ) ` D ) = ( ( (pInvG ` G ) ` B ) ` D ) )
25		hypcgrlem2.b	. . . . . . . . 9 |- ( ph -> B = E )
26	25	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> B = E )
27	1, 2, 3, 14, 15, 5, 11, 16, 21	mirinv 25561	. . . . . . . 8 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( ( (pInvG ` G ) ` B ) ` E ) = E <-> B = E ) )
28	26, 27	mpbird 247	. . . . . . 7 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( (pInvG ` G ) ` B ) ` E ) = E )
29	28	eqcomd 2628	. . . . . 6 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> E = ( ( (pInvG ` G ) ` B ) ` E ) )
30		hypcgr.f	. . . . . . . . . 10 |- ( ph -> F e. P )
31	30	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> F e. P )
32	1, 2, 3, 5, 7, 13, 31	midcom 25674	. . . . . . . 8 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( C (midG ` G ) F ) = ( F (midG ` G ) C ) )
33		simpr 477	. . . . . . . 8 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( C (midG ` G ) F ) = B )
34	32, 33	eqtr3d 2658	. . . . . . 7 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( F (midG ` G ) C ) = B )
35	1, 2, 3, 5, 7, 31, 13, 15, 11	ismidb 25670	. . . . . . 7 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( C = ( ( (pInvG ` G ) ` B ) ` F ) <-> ( F (midG ` G ) C ) = B ) )
36	34, 35	mpbird 247	. . . . . 6 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> C = ( ( (pInvG ` G ) ` B ) ` F ) )
37	24, 29, 36	s3eqd 13609	. . . . 5 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> <" ( ( (pInvG ` G ) ` B ) ` D ) E C "> = <" ( ( (pInvG ` G ) ` B ) ` D ) ( ( (pInvG ` G ) ` B ) ` E ) ( ( (pInvG ` G ) ` B ) ` F ) "> )
38		hypcgr.2	. . . . . . 7 |- ( ph -> <" D E F "> e. (∟G ` G ) )
39	38	adantr 481	. . . . . 6 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> <" D E F "> e. (∟G ` G ) )
40	1, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11	mirrag 25596	. . . . 5 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> <" ( ( (pInvG ` G ) ` B ) ` D ) ( ( (pInvG ` G ) ` B ) ` E ) ( ( (pInvG ` G ) ` B ) ` F ) "> e. (∟G ` G ) )
41	37, 40	eqeltrd 2701	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> <" ( ( (pInvG ` G ) ` B ) ` D ) E C "> e. (∟G ` G ) )
42		hypcgr.3	. . . . . 6 |- ( ph -> ( A .- B ) = ( D .- E ) )
43	42	adantr 481	. . . . 5 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( A .- B ) = ( D .- E ) )
44	1, 2, 3, 14, 15, 5, 11, 16, 18, 21	miriso 25565	. . . . 5 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( ( (pInvG ` G ) ` B ) ` D ) .- ( ( (pInvG ` G ) ` B ) ` E ) ) = ( D .- E ) )
45	28	oveq2d 6666	. . . . 5 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( ( (pInvG ` G ) ` B ) ` D ) .- ( ( (pInvG ` G ) ` B ) ` E ) ) = ( ( ( (pInvG ` G ) ` B ) ` D ) .- E ) )
46	43, 44, 45	3eqtr2d 2662	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( A .- B ) = ( ( ( (pInvG ` G ) ` B ) ` D ) .- E ) )
47	26	oveq1d 6665	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( B .- C ) = ( E .- C ) )
48		eqid 2622	. . . 4 |- ( (lInvG ` G ) ` ( ( A (midG ` G ) ( ( (pInvG ` G ) ` B ) ` D ) ) (LineG ` G ) B ) ) = ( (lInvG ` G ) ` ( ( A (midG ` G ) ( ( (pInvG ` G ) ` B ) ` D ) ) (LineG ` G ) B ) )
49		eqidd 2623	. . . 4 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> C = C )
50	1, 2, 3, 5, 7, 9, 11, 13, 19, 21, 13, 23, 41, 46, 47, 26, 48, 49	hypcgrlem1 25691	. . 3 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( A .- C ) = ( ( ( (pInvG ` G ) ` B ) ` D ) .- C ) )
51	36	oveq2d 6666	. . 3 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( ( (pInvG ` G ) ` B ) ` D ) .- C ) = ( ( ( (pInvG ` G ) ` B ) ` D ) .- ( ( (pInvG ` G ) ` B ) ` F ) ) )
52	1, 2, 3, 14, 15, 5, 11, 16, 18, 31	miriso 25565	. . 3 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( ( ( (pInvG ` G ) ` B ) ` D ) .- ( ( (pInvG ` G ) ` B ) ` F ) ) = ( D .- F ) )
53	50, 51, 52	3eqtrd 2660	. 2 |- ( ( ph /\ ( C (midG ` G ) F ) = B ) -> ( A .- C ) = ( D .- F ) )
54	4	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> G e. TarskiG )
55	6	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> GDimTarskiG≥ 2 )
56	8	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> A e. P )
57	10	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> B e. P )
58	12	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> C e. P )
59	17	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> D e. P )
60	20	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> E e. P )
61	30	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> F e. P )
62	22	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> <" A B C "> e. (∟G ` G ) )
63	38	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> <" D E F "> e. (∟G ` G ) )
64	42	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> ( A .- B ) = ( D .- E ) )
65		hypcgr.4	. . . . 5 |- ( ph -> ( B .- C ) = ( E .- F ) )
66	65	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> ( B .- C ) = ( E .- F ) )
67	25	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> B = E )
68		eqid 2622	. . . 4 |- ( (lInvG ` G ) ` ( ( A (midG ` G ) D ) (LineG ` G ) B ) ) = ( (lInvG ` G ) ` ( ( A (midG ` G ) D ) (LineG ` G ) B ) )
69		simpr 477	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> C = F )
70	1, 2, 3, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69	hypcgrlem1 25691	. . 3 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = F ) -> ( A .- C ) = ( D .- F ) )
71	4	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> G e. TarskiG )
72	6	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> GDimTarskiG≥ 2 )
73	8	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> A e. P )
74	10	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> B e. P )
75	12	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> C e. P )
76		hypcgrlem2.s	. . . . . 6 |- S = ( (lInvG ` G ) ` ( ( C (midG ` G ) F ) (LineG ` G ) B ) )
77	30	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> F e. P )
78	1, 2, 3, 71, 72, 75, 77	midcl 25669	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) e. P )
79		simplr 792	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) =/= B )
80	1, 3, 14, 71, 78, 74, 79	tgelrnln 25525	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( C (midG ` G ) F ) (LineG ` G ) B ) e. ran (LineG ` G ) )
81	17	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> D e. P )
82	1, 2, 3, 71, 72, 76, 14, 80, 81	lmicl 25678	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( S ` D ) e. P )
83	20	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> E e. P )
84	1, 2, 3, 71, 72, 76, 14, 80, 83	lmicl 25678	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( S ` E ) e. P )
85	1, 2, 3, 71, 72, 76, 14, 80, 77	lmicl 25678	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( S ` F ) e. P )
86	22	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> <" A B C "> e. (∟G ` G ) )
87	1, 2, 3, 71, 72, 76, 14, 80	lmimot 25690	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> S e. ( GIsmt G ) )
88	38	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> <" D E F "> e. (∟G ` G ) )
89	1, 2, 3, 14, 15, 71, 81, 83, 77, 87, 88	motrag 25603	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> <" ( S ` D ) ( S ` E ) ( S ` F ) "> e. (∟G ` G ) )
90	42	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( A .- B ) = ( D .- E ) )
91	1, 2, 3, 71, 72, 76, 14, 80, 81, 83	lmiiso 25689	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( S ` D ) .- ( S ` E ) ) = ( D .- E ) )
92	90, 91	eqtr4d 2659	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( A .- B ) = ( ( S ` D ) .- ( S ` E ) ) )
93	65	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( B .- C ) = ( E .- F ) )
94	1, 2, 3, 71, 72, 76, 14, 80, 83, 77	lmiiso 25689	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( S ` E ) .- ( S ` F ) ) = ( E .- F ) )
95	93, 94	eqtr4d 2659	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( B .- C ) = ( ( S ` E ) .- ( S ` F ) ) )
96	1, 3, 14, 71, 78, 74, 79	tglinerflx2 25529	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> B e. ( ( C (midG ` G ) F ) (LineG ` G ) B ) )
97	1, 2, 3, 71, 72, 76, 14, 80, 74	lmiinv 25684	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( S ` B ) = B <-> B e. ( ( C (midG ` G ) F ) (LineG ` G ) B ) ) )
98	96, 97	mpbird 247	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( S ` B ) = B )
99	25	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> B = E )
100	99	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( S ` B ) = ( S ` E ) )
101	98, 100	eqtr3d 2658	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> B = ( S ` E ) )
102		eqid 2622	. . . . 5 |- ( (lInvG ` G ) ` ( ( A (midG ` G ) ( S ` D ) ) (LineG ` G ) B ) ) = ( (lInvG ` G ) ` ( ( A (midG ` G ) ( S ` D ) ) (LineG ` G ) B ) )
103	1, 2, 3, 71, 72, 75, 77	midcom 25674	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) = ( F (midG ` G ) C ) )
104	1, 3, 14, 71, 78, 74, 79	tglinerflx1 25528	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) e. ( ( C (midG ` G ) F ) (LineG ` G ) B ) )
105	103, 104	eqeltrrd 2702	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( F (midG ` G ) C ) e. ( ( C (midG ` G ) F ) (LineG ` G ) B ) )
106		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> C =/= F )
107	106	necomd 2849	. . . . . . . . 9 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> F =/= C )
108	1, 3, 14, 71, 77, 75, 107	tgelrnln 25525	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( F (LineG ` G ) C ) e. ran (LineG ` G ) )
109	1, 2, 3, 71, 72, 75, 77	midbtwn 25671	. . . . . . . . . . 11 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) e. ( C I F ) )
110	1, 2, 3, 71, 75, 78, 77, 109	tgbtwncom 25383	. . . . . . . . . 10 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) e. ( F I C ) )
111	1, 3, 14, 71, 77, 75, 78, 107, 110	btwnlng1 25514	. . . . . . . . 9 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) e. ( F (LineG ` G ) C ) )
112	104, 111	elind 3798	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) e. ( ( ( C (midG ` G ) F ) (LineG ` G ) B ) i^i ( F (LineG ` G ) C ) ) )
113	1, 3, 14, 71, 77, 75, 107	tglinerflx2 25529	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> C e. ( F (LineG ` G ) C ) )
114	79	necomd 2849	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> B =/= ( C (midG ` G ) F ) )
115	4	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> G e. TarskiG )
116	12	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> C e. P )
117	30	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> F e. P )
118	6	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> GDimTarskiG≥ 2 )
119		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> C = ( C (midG ` G ) F ) )
120	119	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> ( C (midG ` G ) F ) = C )
121	1, 2, 3, 115, 118, 116, 117, 120	midcgr 25672	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> ( C .- C ) = ( C .- F ) )
122	121	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> ( C .- F ) = ( C .- C ) )
123	1, 2, 3, 115, 116, 117, 116, 122	axtgcgrid 25362	. . . . . . . . . . 11 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C = ( C (midG ` G ) F ) ) -> C = F )
124	123	ex 450	. . . . . . . . . 10 |- ( ( ph /\ ( C (midG ` G ) F ) =/= B ) -> ( C = ( C (midG ` G ) F ) -> C = F ) )
125	124	necon3d 2815	. . . . . . . . 9 |- ( ( ph /\ ( C (midG ` G ) F ) =/= B ) -> ( C =/= F -> C =/= ( C (midG ` G ) F ) ) )
126	125	imp 445	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> C =/= ( C (midG ` G ) F ) )
127	99	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> E = B )
128		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C (midG ` G ) F ) = ( C (midG ` G ) F ) )
129	1, 2, 3, 71, 72, 75, 77, 15, 78	ismidb 25670	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( F = ( ( (pInvG ` G ) ` ( C (midG ` G ) F ) ) ` C ) <-> ( C (midG ` G ) F ) = ( C (midG ` G ) F ) ) )
130	128, 129	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> F = ( ( (pInvG ` G ) ` ( C (midG ` G ) F ) ) ` C ) )
131	127, 130	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( E .- F ) = ( B .- ( ( (pInvG ` G ) ` ( C (midG ` G ) F ) ) ` C ) ) )
132	93, 131	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( B .- C ) = ( B .- ( ( (pInvG ` G ) ` ( C (midG ` G ) F ) ) ` C ) ) )
133	1, 2, 3, 14, 15, 71, 74, 78, 75	israg 25592	. . . . . . . . 9 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( <" B ( C (midG ` G ) F ) C "> e. (∟G ` G ) <-> ( B .- C ) = ( B .- ( ( (pInvG ` G ) ` ( C (midG ` G ) F ) ) ` C ) ) ) )
134	132, 133	mpbird 247	. . . . . . . 8 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> <" B ( C (midG ` G ) F ) C "> e. (∟G ` G ) )
135	1, 2, 3, 14, 71, 80, 108, 112, 96, 113, 114, 126, 134	ragperp 25612	. . . . . . 7 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( C (midG ` G ) F ) (LineG ` G ) B ) (⟂G ` G ) ( F (LineG ` G ) C ) )
136	135	orcd 407	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( ( C (midG ` G ) F ) (LineG ` G ) B ) (⟂G ` G ) ( F (LineG ` G ) C ) \/ F = C ) )
137	1, 2, 3, 71, 72, 76, 14, 80, 77, 75	islmib 25679	. . . . . 6 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( C = ( S ` F ) <-> ( ( F (midG ` G ) C ) e. ( ( C (midG ` G ) F ) (LineG ` G ) B ) /\ ( ( ( C (midG ` G ) F ) (LineG ` G ) B ) (⟂G ` G ) ( F (LineG ` G ) C ) \/ F = C ) ) ) )
138	105, 136, 137	mpbir2and 957	. . . . 5 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> C = ( S ` F ) )
139	1, 2, 3, 71, 72, 73, 74, 75, 82, 84, 85, 86, 89, 92, 95, 101, 102, 138	hypcgrlem1 25691	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( A .- C ) = ( ( S ` D ) .- ( S ` F ) ) )
140	1, 2, 3, 71, 72, 76, 14, 80, 81, 77	lmiiso 25689	. . . 4 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( ( S ` D ) .- ( S ` F ) ) = ( D .- F ) )
141	139, 140	eqtrd 2656	. . 3 |- ( ( ( ph /\ ( C (midG ` G ) F ) =/= B ) /\ C =/= F ) -> ( A .- C ) = ( D .- F ) )
142	70, 141	pm2.61dane 2881	. 2 |- ( ( ph /\ ( C (midG ` G ) F ) =/= B ) -> ( A .- C ) = ( D .- F ) )
143	53, 142	pm2.61dane 2881	1 |- ( ph -> ( A .- C ) = ( D .- F ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   2c2 11070   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂Gcperpg 25590  midGcmid 25664  lInvGclmi 25665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-ismt 25428  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591  df-mid 25666  df-lmi 25667

This theorem is referenced by:  hypcgr  25693