Theorem tgasa1 25739

Description: Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)

Hypotheses

Ref	Expression
tgsas.p	|- P = ( Base ` G )
tgsas.m	|- .- = ( dist ` G )
tgsas.i	|- I = (Itv ` G )
tgsas.g	|- ( ph -> G e. TarskiG )
tgsas.a	|- ( ph -> A e. P )
tgsas.b	|- ( ph -> B e. P )
tgsas.c	|- ( ph -> C e. P )
tgsas.d	|- ( ph -> D e. P )
tgsas.e	|- ( ph -> E e. P )
tgsas.f	|- ( ph -> F e. P )
tgasa.l	|- L = (LineG ` G )
tgasa.1	|- ( ph -> -. ( C e. ( A L B ) \/ A = B ) )
tgasa.2	|- ( ph -> ( A .- B ) = ( D .- E ) )
tgasa.3	|- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
tgasa.4	|- ( ph -> <" C A B "> (cgrA ` G ) <" F D E "> )

Assertion

Ref	Expression
tgasa1	|- ( ph -> ( B .- C ) = ( E .- F ) )

Proof of Theorem tgasa1

Dummy variables a b f w t u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 796	. . 3 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( E .- f ) = ( B .- C ) )
2		tgsas.p	. . . . 5 |- P = ( Base ` G )
3		tgsas.i	. . . . 5 |- I = (Itv ` G )
4		tgasa.l	. . . . 5 |- L = (LineG ` G )
5		tgsas.g	. . . . . 6 |- ( ph -> G e. TarskiG )
6	5	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> G e. TarskiG )
7		tgsas.f	. . . . . 6 |- ( ph -> F e. P )
8	7	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> F e. P )
9		tgsas.d	. . . . . 6 |- ( ph -> D e. P )
10	9	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> D e. P )
11		tgsas.e	. . . . . 6 |- ( ph -> E e. P )
12	11	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> E e. P )
13		tgsas.m	. . . . . . 7 |- .- = ( dist ` G )
14		tgsas.a	. . . . . . 7 |- ( ph -> A e. P )
15		tgsas.b	. . . . . . 7 |- ( ph -> B e. P )
16		tgsas.c	. . . . . . 7 |- ( ph -> C e. P )
17		tgasa.3	. . . . . . 7 |- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
18		tgasa.1	. . . . . . 7 |- ( ph -> -. ( C e. ( A L B ) \/ A = B ) )
19	2, 3, 13, 5, 14, 15, 16, 9, 11, 7, 17, 4, 18	cgrancol 25720	. . . . . 6 |- ( ph -> -. ( F e. ( D L E ) \/ D = E ) )
20	19	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> -. ( F e. ( D L E ) \/ D = E ) )
21		eqid 2622	. . . . . 6 |- (hlG ` G ) = (hlG ` G )
22		simplr 792	. . . . . 6 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f e. P )
23	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> C e. P )
24	14	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> A e. P )
25	15	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> B e. P )
26	18	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> -. ( C e. ( A L B ) \/ A = B ) )
27	6	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> G e. TarskiG )
28	10	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> D e. P )
29	12	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> E e. P )
30	8	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> F e. P )
31	24	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> A e. P )
32	25	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> B e. P )
33	23	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> C e. P )
34	2, 3, 5, 21, 14, 15, 16, 9, 11, 7, 17	cgracom 25714	. . . . . . . . . 10 |- ( ph -> <" D E F "> (cgrA ` G ) <" A B C "> )
35	34	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> <" D E F "> (cgrA ` G ) <" A B C "> )
36		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> ( E e. ( D L F ) \/ D = F ) )
37	2, 4, 3, 27, 28, 30, 29, 36	colcom 25453	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> ( E e. ( F L D ) \/ F = D ) )
38	2, 4, 3, 27, 30, 28, 29, 37	colrot1 25454	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> ( F e. ( D L E ) \/ D = E ) )
39	2, 3, 13, 27, 28, 29, 30, 31, 32, 33, 35, 4, 38	cgracol 25719	. . . . . . . 8 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> ( C e. ( A L B ) \/ A = B ) )
40	26	adantr 481	. . . . . . . 8 |- ( ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) /\ ( E e. ( D L F ) \/ D = F ) ) -> -. ( C e. ( A L B ) \/ A = B ) )
41	39, 40	pm2.65da 600	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> -. ( E e. ( D L F ) \/ D = F ) )
42		eqid 2622	. . . . . . . . . 10 |- (cgrG ` G ) = (cgrG ` G )
43	17	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" A B C "> (cgrA ` G ) <" D E F "> )
44		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f ( (hlG ` G ) ` E ) F )
45	2, 3, 21, 6, 24, 25, 23, 10, 12, 8, 43, 22, 44	cgrahl2 25709	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" A B C "> (cgrA ` G ) <" D E f "> )
46	2, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17	cgrane1 25704	. . . . . . . . . . . . . 14 |- ( ph -> A =/= B )
47	2, 3, 21, 14, 14, 15, 5, 46	hlid 25504	. . . . . . . . . . . . 13 |- ( ph -> A ( (hlG ` G ) ` B ) A )
48	47	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> A ( (hlG ` G ) ` B ) A )
49	2, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17	cgrane2 25705	. . . . . . . . . . . . . . 15 |- ( ph -> B =/= C )
50	49	necomd 2849	. . . . . . . . . . . . . 14 |- ( ph -> C =/= B )
51	2, 3, 21, 16, 14, 15, 5, 50	hlid 25504	. . . . . . . . . . . . 13 |- ( ph -> C ( (hlG ` G ) ` B ) C )
52	51	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> C ( (hlG ` G ) ` B ) C )
53		tgasa.2	. . . . . . . . . . . . . 14 |- ( ph -> ( A .- B ) = ( D .- E ) )
54	2, 13, 3, 5, 14, 15, 9, 11, 53	tgcgrcomlr 25375	. . . . . . . . . . . . 13 |- ( ph -> ( B .- A ) = ( E .- D ) )
55	54	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( B .- A ) = ( E .- D ) )
56	1	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( B .- C ) = ( E .- f ) )
57	2, 3, 21, 6, 24, 25, 23, 10, 12, 22, 45, 24, 13, 23, 48, 52, 55, 56	cgracgr 25710	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( A .- C ) = ( D .- f ) )
58	2, 13, 3, 6, 24, 23, 10, 22, 57	tgcgrcomlr 25375	. . . . . . . . . 10 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( C .- A ) = ( f .- D ) )
59	53	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( A .- B ) = ( D .- E ) )
60	2, 13, 42, 6, 23, 24, 25, 22, 10, 12, 58, 59, 56	trgcgr 25411	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" C A B "> (cgrG ` G ) <" f D E "> )
61	2, 3, 4, 5, 16, 14, 15, 18	ncolne1 25520	. . . . . . . . . . . 12 |- ( ph -> C =/= A )
62	61	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> C =/= A )
63	2, 13, 3, 6, 23, 24, 22, 10, 58, 62	tgcgrneq 25378	. . . . . . . . . 10 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f =/= D )
64	2, 3, 21, 22, 8, 10, 6, 63	hlid 25504	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f ( (hlG ` G ) ` D ) f )
65	2, 3, 21, 5, 9, 11, 7, 14, 15, 16, 34	cgrane1 25704	. . . . . . . . . . . 12 |- ( ph -> D =/= E )
66	65	necomd 2849	. . . . . . . . . . 11 |- ( ph -> E =/= D )
67	2, 3, 21, 11, 14, 9, 5, 66	hlid 25504	. . . . . . . . . 10 |- ( ph -> E ( (hlG ` G ) ` D ) E )
68	67	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> E ( (hlG ` G ) ` D ) E )
69	2, 3, 21, 6, 23, 24, 25, 22, 10, 12, 22, 12, 60, 64, 68	iscgrad 25703	. . . . . . . 8 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" C A B "> (cgrA ` G ) <" f D E "> )
70	65	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> D =/= E )
71	2, 3, 6, 21, 22, 10, 12, 63, 70	cgraswap 25712	. . . . . . . 8 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" f D E "> (cgrA ` G ) <" E D f "> )
72	2, 3, 6, 21, 23, 24, 25, 22, 10, 12, 69, 12, 10, 22, 71	cgratr 25715	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" C A B "> (cgrA ` G ) <" E D f "> )
73		tgasa.4	. . . . . . . . 9 |- ( ph -> <" C A B "> (cgrA ` G ) <" F D E "> )
74	2, 3, 21, 5, 16, 14, 15, 7, 9, 11, 73	cgrane3 25706	. . . . . . . . . . 11 |- ( ph -> D =/= F )
75	74	necomd 2849	. . . . . . . . . 10 |- ( ph -> F =/= D )
76	2, 3, 5, 21, 7, 9, 11, 75, 65	cgraswap 25712	. . . . . . . . 9 |- ( ph -> <" F D E "> (cgrA ` G ) <" E D F "> )
77	2, 3, 5, 21, 16, 14, 15, 7, 9, 11, 73, 11, 9, 7, 76	cgratr 25715	. . . . . . . 8 |- ( ph -> <" C A B "> (cgrA ` G ) <" E D F "> )
78	77	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> <" C A B "> (cgrA ` G ) <" E D F "> )
79	2, 3, 4, 5, 11, 9, 66	tgelrnln 25525	. . . . . . . . 9 |- ( ph -> ( E L D ) e. ran L )
80	79	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( E L D ) e. ran L )
81		simpl 473	. . . . . . . . . . . 12 |- ( ( a = u /\ b = v ) -> a = u )
82		eqidd 2623	. . . . . . . . . . . 12 |- ( ( a = u /\ b = v ) -> ( P \ ( E L D ) ) = ( P \ ( E L D ) ) )
83	81, 82	eleq12d 2695	. . . . . . . . . . 11 |- ( ( a = u /\ b = v ) -> ( a e. ( P \ ( E L D ) ) <-> u e. ( P \ ( E L D ) ) ) )
84		simpr 477	. . . . . . . . . . . 12 |- ( ( a = u /\ b = v ) -> b = v )
85	84, 82	eleq12d 2695	. . . . . . . . . . 11 |- ( ( a = u /\ b = v ) -> ( b e. ( P \ ( E L D ) ) <-> v e. ( P \ ( E L D ) ) ) )
86	83, 85	anbi12d 747	. . . . . . . . . 10 |- ( ( a = u /\ b = v ) -> ( ( a e. ( P \ ( E L D ) ) /\ b e. ( P \ ( E L D ) ) ) <-> ( u e. ( P \ ( E L D ) ) /\ v e. ( P \ ( E L D ) ) ) ) )
87		simpr 477	. . . . . . . . . . . 12 |- ( ( ( a = u /\ b = v ) /\ t = w ) -> t = w )
88		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( a = u /\ b = v ) /\ t = w ) -> a = u )
89		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( a = u /\ b = v ) /\ t = w ) -> b = v )
90	88, 89	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( a = u /\ b = v ) /\ t = w ) -> ( a I b ) = ( u I v ) )
91	87, 90	eleq12d 2695	. . . . . . . . . . 11 |- ( ( ( a = u /\ b = v ) /\ t = w ) -> ( t e. ( a I b ) <-> w e. ( u I v ) ) )
92	91	cbvrexdva 3178	. . . . . . . . . 10 |- ( ( a = u /\ b = v ) -> ( E. t e. ( E L D ) t e. ( a I b ) <-> E. w e. ( E L D ) w e. ( u I v ) ) )
93	86, 92	anbi12d 747	. . . . . . . . 9 |- ( ( a = u /\ b = v ) -> ( ( ( a e. ( P \ ( E L D ) ) /\ b e. ( P \ ( E L D ) ) ) /\ E. t e. ( E L D ) t e. ( a I b ) ) <-> ( ( u e. ( P \ ( E L D ) ) /\ v e. ( P \ ( E L D ) ) ) /\ E. w e. ( E L D ) w e. ( u I v ) ) ) )
94	93	cbvopabv 4722	. . . . . . . 8 |- { <. a , b >. | ( ( a e. ( P \ ( E L D ) ) /\ b e. ( P \ ( E L D ) ) ) /\ E. t e. ( E L D ) t e. ( a I b ) ) } = { <. u , v >. | ( ( u e. ( P \ ( E L D ) ) /\ v e. ( P \ ( E L D ) ) ) /\ E. w e. ( E L D ) w e. ( u I v ) ) }
95	2, 3, 4, 5, 11, 9, 66	tglinerflx1 25528	. . . . . . . . . 10 |- ( ph -> E e. ( E L D ) )
96	95	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> E e. ( E L D ) )
97	2, 4, 3, 5, 9, 11, 7, 19	ncolcom 25456	. . . . . . . . . . 11 |- ( ph -> -. ( F e. ( E L D ) \/ E = D ) )
98		pm2.45 412	. . . . . . . . . . 11 |- ( -. ( F e. ( E L D ) \/ E = D ) -> -. F e. ( E L D ) )
99	97, 98	syl 17	. . . . . . . . . 10 |- ( ph -> -. F e. ( E L D ) )
100	99	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> -. F e. ( E L D ) )
101	2, 3, 21, 22, 8, 12, 6, 44	hlcomd 25499	. . . . . . . . 9 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> F ( (hlG ` G ) ` E ) f )
102	2, 3, 4, 6, 80, 12, 94, 21, 96, 8, 22, 100, 101	hphl 25663	. . . . . . . 8 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> F ( (hpG ` G ) ` ( E L D ) ) f )
103	2, 3, 4, 6, 80, 8, 94, 22, 102	hpgcom 25659	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f ( (hpG ` G ) ` ( E L D ) ) F )
104	2, 3, 4, 5, 79, 7, 94, 99	hpgid 25658	. . . . . . . 8 |- ( ph -> F ( (hpG ` G ) ` ( E L D ) ) F )
105	104	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> F ( (hpG ` G ) ` ( E L D ) ) F )
106	2, 3, 13, 6, 23, 24, 25, 12, 10, 8, 4, 26, 41, 22, 8, 21, 72, 78, 103, 105	acopyeu 25725	. . . . . 6 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f ( (hlG ` G ) ` D ) F )
107	2, 3, 21, 22, 8, 10, 6, 4, 106	hlln 25502	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f e. ( F L D ) )
108	2, 3, 4, 5, 7, 9, 75	tglinerflx1 25528	. . . . . 6 |- ( ph -> F e. ( F L D ) )
109	108	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> F e. ( F L D ) )
110	2, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17	cgrane4 25707	. . . . . . 7 |- ( ph -> E =/= F )
111	110	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> E =/= F )
112	2, 3, 21, 22, 8, 12, 6, 4, 44	hlln 25502	. . . . . 6 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f e. ( F L E ) )
113	2, 3, 4, 6, 12, 8, 22, 111, 112	lncom 25517	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f e. ( E L F ) )
114	2, 3, 4, 6, 12, 8, 111	tglinerflx2 25529	. . . . 5 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> F e. ( E L F ) )
115	2, 3, 4, 6, 8, 10, 12, 8, 20, 107, 109, 113, 114	tglineinteq 25540	. . . 4 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> f = F )
116	115	oveq2d 6666	. . 3 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( E .- f ) = ( E .- F ) )
117	1, 116	eqtr3d 2658	. 2 |- ( ( ( ph /\ f e. P ) /\ ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) ) -> ( B .- C ) = ( E .- F ) )
118	110	necomd 2849	. . 3 |- ( ph -> F =/= E )
119	2, 3, 21, 11, 15, 16, 5, 7, 13, 118, 49	hlcgrex 25511	. 2 |- ( ph -> E. f e. P ( f ( (hlG ` G ) ` E ) F /\ ( E .- f ) = ( B .- C ) ) )
120	117, 119	r19.29a 3078	1 |- ( ph -> ( B .- C ) = ( E .- F ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  hlGchlg 25495  hpGchpg 25649  cgrAccgra 25699

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-fal 1489  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-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591  df-hpg 25650  df-mid 25666  df-lmi 25667  df-cgra 25700

This theorem is referenced by:  tgasa  25740