Theorem hypcgrlem1 25691

Description: Lemma for hypcgr 25693, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-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 )
hypcgrlem1.s	|- S = ( (lInvG ` G ) ` ( ( A (midG ` G ) D ) (LineG ` G ) B ) )
hypcgrlem1.a	|- ( ph -> C = F )

Assertion

Ref	Expression
hypcgrlem1	|- ( ph -> ( A .- C ) = ( D .- F ) )

Proof of Theorem hypcgrlem1

Step	Hyp	Ref	Expression
1		hypcgr.p	. . 3 |- P = ( Base ` G )
2		hypcgr.m	. . 3 |- .- = ( dist ` G )
3		hypcgr.i	. . 3 |- I = (Itv ` G )
4		hypcgr.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . 3 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> G e. TarskiG )
6		hypcgr.c	. . . 4 |- ( ph -> C e. P )
7	6	adantr 481	. . 3 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> C e. P )
8		hypcgr.a	. . . 4 |- ( ph -> A e. P )
9	8	adantr 481	. . 3 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> A e. P )
10		hypcgr.f	. . . 4 |- ( ph -> F e. P )
11	10	adantr 481	. . 3 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> F e. P )
12		hypcgr.d	. . . 4 |- ( ph -> D e. P )
13	12	adantr 481	. . 3 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> D e. P )
14		eqid 2622	. . . . . . 7 |- (LineG ` G ) = (LineG ` G )
15		eqid 2622	. . . . . . 7 |- (pInvG ` G ) = (pInvG ` G )
16		hypcgr.b	. . . . . . 7 |- ( ph -> B e. P )
17		hypcgr.1	. . . . . . 7 |- ( ph -> <" A B C "> e. (∟G ` G ) )
18	1, 2, 3, 14, 15, 4, 8, 16, 6, 17	ragcom 25593	. . . . . 6 |- ( ph -> <" C B A "> e. (∟G ` G ) )
19	1, 2, 3, 14, 15, 4, 6, 16, 8	israg 25592	. . . . . 6 |- ( ph -> ( <" C B A "> e. (∟G ` G ) <-> ( C .- A ) = ( C .- ( ( (pInvG ` G ) ` B ) ` A ) ) ) )
20	18, 19	mpbid 222	. . . . 5 |- ( ph -> ( C .- A ) = ( C .- ( ( (pInvG ` G ) ` B ) ` A ) ) )
21	20	adantr 481	. . . 4 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> ( C .- A ) = ( C .- ( ( (pInvG ` G ) ` B ) ` A ) ) )
22		hypcgrlem1.a	. . . . . . 7 |- ( ph -> C = F )
23	22	eqcomd 2628	. . . . . 6 |- ( ph -> F = C )
24	23	adantr 481	. . . . 5 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> F = C )
25		hypcgr.h	. . . . . . 7 |- ( ph -> GDimTarskiG≥ 2 )
26	1, 2, 3, 4, 25, 8, 12, 15, 16	ismidb 25670	. . . . . 6 |- ( ph -> ( D = ( ( (pInvG ` G ) ` B ) ` A ) <-> ( A (midG ` G ) D ) = B ) )
27	26	biimpar 502	. . . . 5 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> D = ( ( (pInvG ` G ) ` B ) ` A ) )
28	24, 27	oveq12d 6668	. . . 4 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> ( F .- D ) = ( C .- ( ( (pInvG ` G ) ` B ) ` A ) ) )
29	21, 28	eqtr4d 2659	. . 3 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> ( C .- A ) = ( F .- D ) )
30	1, 2, 3, 5, 7, 9, 11, 13, 29	tgcgrcomlr 25375	. 2 |- ( ( ph /\ ( A (midG ` G ) D ) = B ) -> ( A .- C ) = ( D .- F ) )
31		simpr 477	. . . 4 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = D ) -> A = D )
32	22	ad2antrr 762	. . . 4 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = D ) -> C = F )
33	31, 32	oveq12d 6668	. . 3 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = D ) -> ( A .- C ) = ( D .- F ) )
34	17	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> <" A B C "> e. (∟G ` G ) )
35	4	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> G e. TarskiG )
36	8	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> A e. P )
37	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> B e. P )
38	6	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> C e. P )
39	1, 2, 3, 14, 15, 35, 36, 37, 38	israg 25592	. . . . . 6 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( <" A B C "> e. (∟G ` G ) <-> ( A .- C ) = ( A .- ( ( (pInvG ` G ) ` B ) ` C ) ) ) )
40	34, 39	mpbid 222	. . . . 5 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A .- C ) = ( A .- ( ( (pInvG ` G ) ` B ) ` C ) ) )
41	25	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> GDimTarskiG≥ 2 )
42		hypcgrlem1.s	. . . . . . 7 |- S = ( (lInvG ` G ) ` ( ( A (midG ` G ) D ) (LineG ` G ) B ) )
43	12	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> D e. P )
44	1, 2, 3, 35, 41, 36, 43	midcl 25669	. . . . . . . 8 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) e. P )
45		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) =/= B )
46	1, 3, 14, 35, 44, 37, 45	tgelrnln 25525	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( ( A (midG ` G ) D ) (LineG ` G ) B ) e. ran (LineG ` G ) )
47		eqid 2622	. . . . . . 7 |- ( (pInvG ` G ) ` B ) = ( (pInvG ` G ) ` B )
48		eqid 2622	. . . . . . . . 9 |- (cgrG ` G ) = (cgrG ` G )
49	1, 2, 3, 14, 15, 35, 37, 47, 38	mircl 25556	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( ( (pInvG ` G ) ` B ) ` C ) e. P )
50		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> A =/= D )
51	1, 2, 3, 35, 41, 36, 43	midbtwn 25671	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) e. ( A I D ) )
52	1, 14, 3, 35, 36, 44, 43, 51	btwncolg3 25452	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D e. ( A (LineG ` G ) ( A (midG ` G ) D ) ) \/ A = ( A (midG ` G ) D ) ) )
53		eqidd 2623	. . . . . . . . . . . . 13 |- ( ph -> D = D )
54		hypcgrlem2.b	. . . . . . . . . . . . 13 |- ( ph -> B = E )
55	53, 54, 22	s3eqd 13609	. . . . . . . . . . . 12 |- ( ph -> <" D B C "> = <" D E F "> )
56	55	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> <" D B C "> = <" D E F "> )
57		hypcgr.2	. . . . . . . . . . . 12 |- ( ph -> <" D E F "> e. (∟G ` G ) )
58	57	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> <" D E F "> e. (∟G ` G ) )
59	56, 58	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> <" D B C "> e. (∟G ` G ) )
60	1, 2, 3, 14, 15, 35, 43, 37, 38	israg 25592	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( <" D B C "> e. (∟G ` G ) <-> ( D .- C ) = ( D .- ( ( (pInvG ` G ) ` B ) ` C ) ) ) )
61	59, 60	mpbid 222	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D .- C ) = ( D .- ( ( (pInvG ` G ) ` B ) ` C ) ) )
62	1, 14, 3, 35, 36, 43, 44, 48, 38, 49, 2, 50, 52, 40, 61	lncgr 25464	. . . . . . . 8 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( ( A (midG ` G ) D ) .- C ) = ( ( A (midG ` G ) D ) .- ( ( (pInvG ` G ) ` B ) ` C ) ) )
63	1, 2, 3, 14, 15, 35, 44, 37, 38	israg 25592	. . . . . . . 8 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( <" ( A (midG ` G ) D ) B C "> e. (∟G ` G ) <-> ( ( A (midG ` G ) D ) .- C ) = ( ( A (midG ` G ) D ) .- ( ( (pInvG ` G ) ` B ) ` C ) ) ) )
64	62, 63	mpbird 247	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> <" ( A (midG ` G ) D ) B C "> e. (∟G ` G ) )
65	1, 3, 14, 35, 44, 37, 45	tglinerflx1 25528	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) e. ( ( A (midG ` G ) D ) (LineG ` G ) B ) )
66	1, 3, 14, 35, 44, 37, 45	tglinerflx2 25529	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> B e. ( ( A (midG ` G ) D ) (LineG ` G ) B ) )
67	1, 2, 3, 35, 41, 42, 14, 46, 44, 47, 64, 65, 66, 38, 45	lmimid 25686	. . . . . 6 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( S ` C ) = ( ( (pInvG ` G ) ` B ) ` C ) )
68	67	oveq2d 6666	. . . . 5 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A .- ( S ` C ) ) = ( A .- ( ( (pInvG ` G ) ` B ) ` C ) ) )
69	40, 68	eqtr4d 2659	. . . 4 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A .- C ) = ( A .- ( S ` C ) ) )
70	1, 2, 3, 35, 41, 43, 36	midcom 25674	. . . . . . . 8 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D (midG ` G ) A ) = ( A (midG ` G ) D ) )
71	70, 65	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D (midG ` G ) A ) e. ( ( A (midG ` G ) D ) (LineG ` G ) B ) )
72	50	necomd 2849	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> D =/= A )
73	1, 3, 14, 35, 43, 36, 72	tgelrnln 25525	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D (LineG ` G ) A ) e. ran (LineG ` G ) )
74	1, 2, 3, 35, 36, 44, 43, 51	tgbtwncom 25383	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) e. ( D I A ) )
75	1, 3, 14, 35, 43, 36, 44, 72, 74	btwnlng1 25514	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) e. ( D (LineG ` G ) A ) )
76	65, 75	elind 3798	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) e. ( ( ( A (midG ` G ) D ) (LineG ` G ) B ) i^i ( D (LineG ` G ) A ) ) )
77	1, 3, 14, 35, 43, 36, 72	tglinerflx2 25529	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> A e. ( D (LineG ` G ) A ) )
78	45	necomd 2849	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> B =/= ( A (midG ` G ) D ) )
79	4	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> G e. TarskiG )
80	8	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> A e. P )
81	12	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> D e. P )
82	25	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> GDimTarskiG≥ 2 )
83		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> A = ( A (midG ` G ) D ) )
84	83	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> ( A (midG ` G ) D ) = A )
85	1, 2, 3, 79, 82, 80, 81, 84	midcgr 25672	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> ( A .- A ) = ( A .- D ) )
86	85	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> ( A .- D ) = ( A .- A ) )
87	1, 2, 3, 79, 80, 81, 80, 86	axtgcgrid 25362	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A = ( A (midG ` G ) D ) ) -> A = D )
88	87	ex 450	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) D ) =/= B ) -> ( A = ( A (midG ` G ) D ) -> A = D ) )
89	88	necon3d 2815	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) D ) =/= B ) -> ( A =/= D -> A =/= ( A (midG ` G ) D ) ) )
90	89	imp 445	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> A =/= ( A (midG ` G ) D ) )
91		hypcgr.e	. . . . . . . . . . . . . 14 |- ( ph -> E e. P )
92		hypcgr.3	. . . . . . . . . . . . . 14 |- ( ph -> ( A .- B ) = ( D .- E ) )
93	1, 2, 3, 4, 8, 16, 12, 91, 92	tgcgrcomlr 25375	. . . . . . . . . . . . 13 |- ( ph -> ( B .- A ) = ( E .- D ) )
94	54	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ph -> ( B .- D ) = ( E .- D ) )
95	93, 94	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ph -> ( B .- A ) = ( B .- D ) )
96	95	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( B .- A ) = ( B .- D ) )
97		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A (midG ` G ) D ) = ( A (midG ` G ) D ) )
98	1, 2, 3, 35, 41, 36, 43, 15, 44	ismidb 25670	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D = ( ( (pInvG ` G ) ` ( A (midG ` G ) D ) ) ` A ) <-> ( A (midG ` G ) D ) = ( A (midG ` G ) D ) ) )
99	97, 98	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> D = ( ( (pInvG ` G ) ` ( A (midG ` G ) D ) ) ` A ) )
100	99	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( B .- D ) = ( B .- ( ( (pInvG ` G ) ` ( A (midG ` G ) D ) ) ` A ) ) )
101	96, 100	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( B .- A ) = ( B .- ( ( (pInvG ` G ) ` ( A (midG ` G ) D ) ) ` A ) ) )
102	1, 2, 3, 14, 15, 35, 37, 44, 36	israg 25592	. . . . . . . . . 10 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( <" B ( A (midG ` G ) D ) A "> e. (∟G ` G ) <-> ( B .- A ) = ( B .- ( ( (pInvG ` G ) ` ( A (midG ` G ) D ) ) ` A ) ) ) )
103	101, 102	mpbird 247	. . . . . . . . 9 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> <" B ( A (midG ` G ) D ) A "> e. (∟G ` G ) )
104	1, 2, 3, 14, 35, 46, 73, 76, 66, 77, 78, 90, 103	ragperp 25612	. . . . . . . 8 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( ( A (midG ` G ) D ) (LineG ` G ) B ) (⟂G ` G ) ( D (LineG ` G ) A ) )
105	104	orcd 407	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( ( ( A (midG ` G ) D ) (LineG ` G ) B ) (⟂G ` G ) ( D (LineG ` G ) A ) \/ D = A ) )
106	1, 2, 3, 35, 41, 42, 14, 46, 43, 36	islmib 25679	. . . . . . 7 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A = ( S ` D ) <-> ( ( D (midG ` G ) A ) e. ( ( A (midG ` G ) D ) (LineG ` G ) B ) /\ ( ( ( A (midG ` G ) D ) (LineG ` G ) B ) (⟂G ` G ) ( D (LineG ` G ) A ) \/ D = A ) ) ) )
107	71, 105, 106	mpbir2and 957	. . . . . 6 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> A = ( S ` D ) )
108	107	oveq1d 6665	. . . . 5 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A .- ( S ` C ) ) = ( ( S ` D ) .- ( S ` C ) ) )
109	1, 2, 3, 35, 41, 42, 14, 46, 43, 38	lmiiso 25689	. . . . 5 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( ( S ` D ) .- ( S ` C ) ) = ( D .- C ) )
110	22	oveq2d 6666	. . . . . 6 |- ( ph -> ( D .- C ) = ( D .- F ) )
111	110	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( D .- C ) = ( D .- F ) )
112	108, 109, 111	3eqtrd 2660	. . . 4 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A .- ( S ` C ) ) = ( D .- F ) )
113	69, 112	eqtrd 2656	. . 3 |- ( ( ( ph /\ ( A (midG ` G ) D ) =/= B ) /\ A =/= D ) -> ( A .- C ) = ( D .- F ) )
114	33, 113	pm2.61dane 2881	. 2 |- ( ( ph /\ ( A (midG ` G ) D ) =/= B ) -> ( A .- C ) = ( D .- F ) )
115	30, 114	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  cgrGccgrg 25405  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-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591  df-mid 25666  df-lmi 25667

This theorem is referenced by:  hypcgrlem2  25692