Theorem colhp 25662

Description: Half-plane relation for colinear points. Theorem 9.19 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.)

Hypotheses

Ref	Expression
hpgid.p	|- P = ( Base ` G )
hpgid.i	|- I = (Itv ` G )
hpgid.l	|- L = (LineG ` G )
hpgid.g	|- ( ph -> G e. TarskiG )
hpgid.d	|- ( ph -> D e. ran L )
hpgid.a	|- ( ph -> A e. P )
hpgid.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
colopp.b	|- ( ph -> B e. P )
colopp.p	|- ( ph -> C e. D )
colopp.1	|- ( ph -> ( C e. ( A L B ) \/ A = B ) )
colhp.k	|- K = (hlG ` G )

Assertion

Ref	Expression
colhp	|- ( ph -> ( A ( (hpG ` G ) ` D ) B <-> ( A ( K ` C ) B /\ -. A e. D ) ) )

Distinct variable groups:   t, A   t, B   D, a, b, t   G, a, b, t   I, a, b, t   O, a, b, t   P, a, b, t   ph, t   t, C   L, a, b, t   A, a, b   C, a, b

Allowed substitution hints:   ph( a, b)   B( a, b)   K( t, a, b)

Proof of Theorem colhp

Step	Hyp	Ref	Expression
1		ancom 466	. . 3 |- ( ( A ( K ` C ) B /\ -. A e. D ) <-> ( -. A e. D /\ A ( K ` C ) B ) )
2	1	a1i 11	. 2 |- ( ph -> ( ( A ( K ` C ) B /\ -. A e. D ) <-> ( -. A e. D /\ A ( K ` C ) B ) ) )
3		hpgid.p	. . . . 5 |- P = ( Base ` G )
4		hpgid.i	. . . . 5 |- I = (Itv ` G )
5		hpgid.l	. . . . 5 |- L = (LineG ` G )
6		hpgid.g	. . . . . 6 |- ( ph -> G e. TarskiG )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ -. A e. D ) -> G e. TarskiG )
8		hpgid.d	. . . . . 6 |- ( ph -> D e. ran L )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ -. A e. D ) -> D e. ran L )
10		colopp.b	. . . . . 6 |- ( ph -> B e. P )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ -. A e. D ) -> B e. P )
12		hpgid.o	. . . . 5 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
13		eqid 2622	. . . . . . 7 |- ( dist ` G ) = ( dist ` G )
14		eqid 2622	. . . . . . 7 |- (pInvG ` G ) = (pInvG ` G )
15		colopp.p	. . . . . . . 8 |- ( ph -> C e. D )
16	3, 5, 4, 6, 8, 15	tglnpt 25444	. . . . . . 7 |- ( ph -> C e. P )
17		eqid 2622	. . . . . . 7 |- ( (pInvG ` G ) ` C ) = ( (pInvG ` G ) ` C )
18		hpgid.a	. . . . . . 7 |- ( ph -> A e. P )
19	3, 13, 4, 5, 14, 6, 16, 17, 18	mircl 25556	. . . . . 6 |- ( ph -> ( ( (pInvG ` G ) ` C ) ` A ) e. P )
20	19	adantr 481	. . . . 5 |- ( ( ph /\ -. A e. D ) -> ( ( (pInvG ` G ) ` C ) ` A ) e. P )
21	15	adantr 481	. . . . 5 |- ( ( ph /\ -. A e. D ) -> C e. D )
22	16	adantr 481	. . . . . 6 |- ( ( ph /\ -. A e. D ) -> C e. P )
23	18	adantr 481	. . . . . . 7 |- ( ( ph /\ -. A e. D ) -> A e. P )
24		nelne2 2891	. . . . . . . . . . 11 |- ( ( C e. D /\ -. A e. D ) -> C =/= A )
25	15, 24	sylan 488	. . . . . . . . . 10 |- ( ( ph /\ -. A e. D ) -> C =/= A )
26	25	necomd 2849	. . . . . . . . 9 |- ( ( ph /\ -. A e. D ) -> A =/= C )
27	26	neneqd 2799	. . . . . . . 8 |- ( ( ph /\ -. A e. D ) -> -. A = C )
28		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. A e. D ) -> -. A e. D )
29	3, 13, 4, 5, 14, 6, 16, 17, 18	mirmir 25557	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( (pInvG ` G ) ` C ) ` ( ( (pInvG ` G ) ` C ) ` A ) ) = A )
30	29	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> ( ( (pInvG ` G ) ` C ) ` ( ( (pInvG ` G ) ` C ) ` A ) ) = A )
31	6	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> G e. TarskiG )
32	8	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> D e. ran L )
33	15	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> C e. D )
34		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> ( ( (pInvG ` G ) ` C ) ` A ) e. D )
35	3, 13, 4, 5, 14, 31, 17, 32, 33, 34	mirln 25571	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> ( ( (pInvG ` G ) ` C ) ` ( ( (pInvG ` G ) ` C ) ` A ) ) e. D )
36	30, 35	eqeltrrd 2702	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( ( (pInvG ` G ) ` C ) ` A ) e. D ) -> A e. D )
37	36	stoic1a 1697	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. A e. D ) -> -. ( ( (pInvG ` G ) ` C ) ` A ) e. D )
38		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ t = C ) -> t = C )
39		eqidd 2623	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ t = C ) -> ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) = ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) )
40	38, 39	eleq12d 2695	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ t = C ) -> ( t e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) <-> C e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) ) )
41	3, 13, 4, 5, 14, 6, 16, 17, 18	mirbtwn 25553	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> C e. ( ( ( (pInvG ` G ) ` C ) ` A ) I A ) )
42	3, 13, 4, 6, 19, 16, 18, 41	tgbtwncom 25383	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> C e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) )
43	15, 40, 42	rspcedvd 3317	. . . . . . . . . . . . . . . . . 18 |- ( ph -> E. t e. D t e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) )
44	43	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. A e. D ) -> E. t e. D t e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) )
45	28, 37, 44	jca31 557	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ -. A e. D ) -> ( ( -. A e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) /\ E. t e. D t e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) ) )
46	3, 13, 4, 12, 23, 20	islnopp 25631	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ -. A e. D ) -> ( A O ( ( (pInvG ` G ) ` C ) ` A ) <-> ( ( -. A e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) /\ E. t e. D t e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) ) ) )
47	45, 46	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. A e. D ) -> A O ( ( (pInvG ` G ) ` C ) ` A ) )
48	3, 13, 4, 12, 5, 9, 7, 23, 20, 47	oppne3 25635	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A e. D ) -> A =/= ( ( (pInvG ` G ) ` C ) ` A ) )
49	42	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A e. D ) -> C e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) )
50	3, 4, 5, 7, 23, 20, 22, 48, 49	btwnlng1 25514	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A e. D ) -> C e. ( A L ( ( (pInvG ` G ) ` C ) ` A ) ) )
51	50	orcd 407	. . . . . . . . . . . 12 |- ( ( ph /\ -. A e. D ) -> ( C e. ( A L ( ( (pInvG ` G ) ` C ) ` A ) ) \/ A = ( ( (pInvG ` G ) ` C ) ` A ) ) )
52	3, 5, 4, 7, 23, 20, 22, 51	colcom 25453	. . . . . . . . . . 11 |- ( ( ph /\ -. A e. D ) -> ( C e. ( ( ( (pInvG ` G ) ` C ) ` A ) L A ) \/ ( ( (pInvG ` G ) ` C ) ` A ) = A ) )
53	3, 5, 4, 7, 20, 23, 22, 52	colrot1 25454	. . . . . . . . . 10 |- ( ( ph /\ -. A e. D ) -> ( ( ( (pInvG ` G ) ` C ) ` A ) e. ( A L C ) \/ A = C ) )
54	53	orcomd 403	. . . . . . . . 9 |- ( ( ph /\ -. A e. D ) -> ( A = C \/ ( ( (pInvG ` G ) ` C ) ` A ) e. ( A L C ) ) )
55	54	ord 392	. . . . . . . 8 |- ( ( ph /\ -. A e. D ) -> ( -. A = C -> ( ( (pInvG ` G ) ` C ) ` A ) e. ( A L C ) ) )
56	27, 55	mpd 15	. . . . . . 7 |- ( ( ph /\ -. A e. D ) -> ( ( (pInvG ` G ) ` C ) ` A ) e. ( A L C ) )
57		colopp.1	. . . . . . . . . 10 |- ( ph -> ( C e. ( A L B ) \/ A = B ) )
58	3, 5, 4, 6, 18, 10, 16, 57	colrot1 25454	. . . . . . . . 9 |- ( ph -> ( A e. ( B L C ) \/ B = C ) )
59	3, 5, 4, 6, 10, 16, 18, 58	colcom 25453	. . . . . . . 8 |- ( ph -> ( A e. ( C L B ) \/ C = B ) )
60	59	adantr 481	. . . . . . 7 |- ( ( ph /\ -. A e. D ) -> ( A e. ( C L B ) \/ C = B ) )
61	3, 4, 5, 7, 20, 23, 22, 11, 56, 60	coltr 25542	. . . . . 6 |- ( ( ph /\ -. A e. D ) -> ( ( ( (pInvG ` G ) ` C ) ` A ) e. ( C L B ) \/ C = B ) )
62	3, 5, 4, 7, 22, 11, 20, 61	colrot1 25454	. . . . 5 |- ( ( ph /\ -. A e. D ) -> ( C e. ( B L ( ( (pInvG ` G ) ` C ) ` A ) ) \/ B = ( ( (pInvG ` G ) ` C ) ` A ) ) )
63	3, 4, 5, 7, 9, 11, 12, 20, 21, 62	colopp 25661	. . . 4 |- ( ( ph /\ -. A e. D ) -> ( B O ( ( (pInvG ` G ) ` C ) ` A ) <-> ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) ) )
64	3, 4, 5, 12, 7, 9, 23, 11, 20, 47	lnopp2hpgb 25655	. . . 4 |- ( ( ph /\ -. A e. D ) -> ( B O ( ( (pInvG ` G ) ` C ) ` A ) <-> A ( (hpG ` G ) ` D ) B ) )
65		colhp.k	. . . . . . . . 9 |- K = (hlG ` G )
66		simpll 790	. . . . . . . . . 10 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> ph )
67	66, 10	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> B e. P )
68	66, 18	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> A e. P )
69	66, 16	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> C e. P )
70	66, 6	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> G e. TarskiG )
71	66, 15	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> C e. D )
72		simprr 796	. . . . . . . . . . 11 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> -. B e. D )
73		nelne2 2891	. . . . . . . . . . . 12 |- ( ( C e. D /\ -. B e. D ) -> C =/= B )
74	73	necomd 2849	. . . . . . . . . . 11 |- ( ( C e. D /\ -. B e. D ) -> B =/= C )
75	71, 72, 74	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> B =/= C )
76	26	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> A =/= C )
77		simprl 794	. . . . . . . . . 10 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) )
78	3, 13, 4, 5, 14, 70, 17, 65, 69, 67, 68, 68, 75, 76, 77	mirhl2 25576	. . . . . . . . 9 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> B ( K ` C ) A )
79	3, 4, 65, 67, 68, 69, 70, 78	hlcomd 25499	. . . . . . . 8 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> A ( K ` C ) B )
80	28	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> -. A e. D )
81	79, 80	jca 554	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D ) ) -> ( A ( K ` C ) B /\ -. A e. D ) )
82	81	3adantr3 1222	. . . . . 6 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) ) -> ( A ( K ` C ) B /\ -. A e. D ) )
83	82	simpld 475	. . . . 5 |- ( ( ( ph /\ -. A e. D ) /\ ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) ) -> A ( K ` C ) B )
84	23	adantr 481	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> A e. P )
85	11	adantr 481	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> B e. P )
86	20	adantr 481	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> ( ( (pInvG ` G ) ` C ) ` A ) e. P )
87	7	adantr 481	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> G e. TarskiG )
88	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> C e. P )
89		simpr 477	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> A ( K ` C ) B )
90	42	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> C e. ( A I ( ( (pInvG ` G ) ` C ) ` A ) ) )
91	3, 4, 65, 84, 85, 86, 87, 88, 89, 90	btwnhl 25509	. . . . . 6 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) )
92	3, 4, 65, 84, 85, 88, 87, 5, 89	hlln 25502	. . . . . . . . 9 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> A e. ( B L C ) )
93	92	adantr 481	. . . . . . . 8 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> A e. ( B L C ) )
94	87	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> G e. TarskiG )
95	85	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> B e. P )
96	88	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> C e. P )
97	84	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> A e. P )
98	89	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> A ( K ` C ) B )
99	3, 4, 65, 97, 95, 96, 94, 98	hlne2 25501	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> B =/= C )
100	9	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> D e. ran L )
101		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> B e. D )
102	15	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> C e. D )
103	3, 4, 5, 94, 95, 96, 99, 99, 100, 101, 102	tglinethru 25531	. . . . . . . 8 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> D = ( B L C ) )
104	93, 103	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> A e. D )
105	28	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) /\ B e. D ) -> -. A e. D )
106	104, 105	pm2.65da 600	. . . . . 6 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> -. B e. D )
107	37	adantr 481	. . . . . 6 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> -. ( ( (pInvG ` G ) ` C ) ` A ) e. D )
108	91, 106, 107	3jca 1242	. . . . 5 |- ( ( ( ph /\ -. A e. D ) /\ A ( K ` C ) B ) -> ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) )
109	83, 108	impbida 877	. . . 4 |- ( ( ph /\ -. A e. D ) -> ( ( C e. ( B I ( ( (pInvG ` G ) ` C ) ` A ) ) /\ -. B e. D /\ -. ( ( (pInvG ` G ) ` C ) ` A ) e. D ) <-> A ( K ` C ) B ) )
110	63, 64, 109	3bitr3d 298	. . 3 |- ( ( ph /\ -. A e. D ) -> ( A ( (hpG ` G ) ` D ) B <-> A ( K ` C ) B ) )
111	110	pm5.32da 673	. 2 |- ( ph -> ( ( -. A e. D /\ A ( (hpG ` G ) ` D ) B ) <-> ( -. A e. D /\ A ( K ` C ) B ) ) )
112		simpr 477	. . . 4 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> A ( (hpG ` G ) ` D ) B )
113	112	adantrl 752	. . 3 |- ( ( ph /\ ( -. A e. D /\ A ( (hpG ` G ) ` D ) B ) ) -> A ( (hpG ` G ) ` D ) B )
114	6	adantr 481	. . . . 5 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> G e. TarskiG )
115	8	adantr 481	. . . . 5 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> D e. ran L )
116	18	adantr 481	. . . . 5 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> A e. P )
117	10	adantr 481	. . . . 5 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> B e. P )
118	3, 4, 5, 12, 114, 115, 116, 117, 112	hpgne1 25653	. . . 4 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> -. A e. D )
119	118, 112	jca 554	. . 3 |- ( ( ph /\ A ( (hpG ` G ) ` D ) B ) -> ( -. A e. D /\ A ( (hpG ` G ) ` D ) B ) )
120	113, 119	impbida 877	. 2 |- ( ph -> ( ( -. A e. D /\ A ( (hpG ` G ) ` D ) B ) <-> A ( (hpG ` G ) ` D ) B ) )
121	2, 111, 120	3bitr2rd 297	1 |- ( ph -> ( A ( (hpG ` G ) ` D ) B <-> ( A ( K ` C ) B /\ -. A e. D ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = 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   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  hlGchlg 25495  pInvGcmir 25547  hpGchpg 25649

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-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591  df-hpg 25650

This theorem is referenced by:  hphl  25663  trgcopy  25696