Theorem opphllem5 25643

Description: Second part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 2-Mar-2020.)

Hypotheses

Ref	Expression
hpg.p	|- P = ( Base ` G )
hpg.d	|- .- = ( dist ` G )
hpg.i	|- I = (Itv ` G )
hpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
opphl.l	|- L = (LineG ` G )
opphl.d	|- ( ph -> D e. ran L )
opphl.g	|- ( ph -> G e. TarskiG )
opphl.k	|- K = (hlG ` G )
opphllem5.n	|- N = ( (pInvG ` G ) ` M )
opphllem5.a	|- ( ph -> A e. P )
opphllem5.c	|- ( ph -> C e. P )
opphllem5.r	|- ( ph -> R e. D )
opphllem5.s	|- ( ph -> S e. D )
opphllem5.m	|- ( ph -> M e. P )
opphllem5.o	|- ( ph -> A O C )
opphllem5.p	|- ( ph -> D (⟂G ` G ) ( A L R ) )
opphllem5.q	|- ( ph -> D (⟂G ` G ) ( C L S ) )
opphllem5.u	|- ( ph -> U e. P )
opphllem5.v	|- ( ph -> V e. P )
opphllem5.1	|- ( ph -> U ( K ` R ) A )
opphllem5.2	|- ( ph -> V ( K ` S ) C )

Assertion

Ref	Expression
opphllem5	|- ( ph -> U O V )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, D   t, R   t, C   t, G   t, L   t, U   t, I   t, K   t, M   t, O   t, N   t, P   t, S   t, V   ph, t   t, .-   t, a, b

Allowed substitution hints:   ph( a, b)   A( a, b)   C( a, b)   R( a, b)   S( a, b)   U( a, b)   G( a, b)   K( a, b)   L( a, b)   M( a, b)   .- ( a, b)   N( a, b)   O( a, b)   V( a, b)

Proof of Theorem opphllem5

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hpg.p	. . . . . . 7 |- P = ( Base ` G )
2		hpg.d	. . . . . . 7 |- .- = ( dist ` G )
3		hpg.i	. . . . . . 7 |- I = (Itv ` G )
4		opphl.l	. . . . . . 7 |- L = (LineG ` G )
5		opphl.g	. . . . . . 7 |- ( ph -> G e. TarskiG )
6		opphl.d	. . . . . . 7 |- ( ph -> D e. ran L )
7		opphl.k	. . . . . . 7 |- K = (hlG ` G )
8		opphllem5.r	. . . . . . 7 |- ( ph -> R e. D )
9		opphllem5.a	. . . . . . 7 |- ( ph -> A e. P )
10		opphllem5.u	. . . . . . 7 |- ( ph -> U e. P )
11		opphllem5.p	. . . . . . . 8 |- ( ph -> D (⟂G ` G ) ( A L R ) )
12	1, 4, 3, 5, 6, 8	tglnpt 25444	. . . . . . . . 9 |- ( ph -> R e. P )
13		opphllem5.1	. . . . . . . . . 10 |- ( ph -> U ( K ` R ) A )
14	1, 3, 7, 10, 9, 12, 5, 13	hlne2 25501	. . . . . . . . 9 |- ( ph -> A =/= R )
15	1, 3, 4, 5, 9, 12, 14	tglinecom 25530	. . . . . . . 8 |- ( ph -> ( A L R ) = ( R L A ) )
16	11, 15	breqtrd 4679	. . . . . . 7 |- ( ph -> D (⟂G ` G ) ( R L A ) )
17	1, 3, 7, 10, 9, 12, 5, 13	hlcomd 25499	. . . . . . 7 |- ( ph -> A ( K ` R ) U )
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17	hlperpnel 25617	. . . . . 6 |- ( ph -> -. U e. D )
19	18	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> -. U e. D )
20		opphllem5.s	. . . . . . 7 |- ( ph -> S e. D )
21		opphllem5.c	. . . . . . 7 |- ( ph -> C e. P )
22		opphllem5.v	. . . . . . 7 |- ( ph -> V e. P )
23		opphllem5.q	. . . . . . . 8 |- ( ph -> D (⟂G ` G ) ( C L S ) )
24	1, 4, 3, 5, 6, 20	tglnpt 25444	. . . . . . . . 9 |- ( ph -> S e. P )
25		opphllem5.2	. . . . . . . . . 10 |- ( ph -> V ( K ` S ) C )
26	1, 3, 7, 22, 21, 24, 5, 25	hlne2 25501	. . . . . . . . 9 |- ( ph -> C =/= S )
27	1, 3, 4, 5, 21, 24, 26	tglinecom 25530	. . . . . . . 8 |- ( ph -> ( C L S ) = ( S L C ) )
28	23, 27	breqtrd 4679	. . . . . . 7 |- ( ph -> D (⟂G ` G ) ( S L C ) )
29	1, 3, 7, 22, 21, 24, 5, 25	hlcomd 25499	. . . . . . 7 |- ( ph -> C ( K ` S ) V )
30	1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 28, 29	hlperpnel 25617	. . . . . 6 |- ( ph -> -. V e. D )
31	30	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> -. V e. D )
32		simplr 792	. . . . . 6 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. D )
33		simpr 477	. . . . . . . 8 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R = t ) -> R = t )
34		eqid 2622	. . . . . . . . 9 |- (pInvG ` G ) = (pInvG ` G )
35	5	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> G e. TarskiG )
36	35	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> G e. TarskiG )
37	21	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> C e. P )
38	37	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> C e. P )
39	12	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. P )
40	39	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. P )
41	6	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> D e. ran L )
42	1, 4, 3, 35, 41, 32	tglnpt 25444	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. P )
43	42	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. P )
44	9	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> A e. P )
45	44	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> A e. P )
46	24	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> S e. P )
47	46	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> S e. P )
48		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R = S )
49	1, 3, 4, 5, 21, 24, 26	tglinerflx2 25529	. . . . . . . . . . . . 13 |- ( ph -> S e. ( C L S ) )
50	49	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> S e. ( C L S ) )
51	48, 50	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( C L S ) )
52	51	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. ( C L S ) )
53	1, 3, 4, 5, 21, 24, 26	tgelrnln 25525	. . . . . . . . . . . . 13 |- ( ph -> ( C L S ) e. ran L )
54	1, 2, 3, 4, 5, 6, 53, 23	perpcom 25608	. . . . . . . . . . . 12 |- ( ph -> ( C L S ) (⟂G ` G ) D )
55	54	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( C L S ) (⟂G ` G ) D )
56		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R =/= t )
57	41	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> D e. ran L )
58	8	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. D )
59	58	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. D )
60	32	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. D )
61	1, 3, 4, 36, 40, 43, 56, 56, 57, 59, 60	tglinethru 25531	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> D = ( R L t ) )
62	55, 61	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( C L S ) (⟂G ` G ) ( R L t ) )
63	1, 2, 3, 4, 36, 38, 47, 52, 43, 62	perprag 25618	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> <" C R t "> e. (∟G ` G ) )
64	1, 3, 4, 5, 9, 12, 14	tglinerflx2 25529	. . . . . . . . . . . 12 |- ( ph -> R e. ( A L R ) )
65	64	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( A L R ) )
66	65	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. ( A L R ) )
67	1, 3, 4, 5, 9, 12, 14	tgelrnln 25525	. . . . . . . . . . . . 13 |- ( ph -> ( A L R ) e. ran L )
68	1, 2, 3, 4, 5, 6, 67, 11	perpcom 25608	. . . . . . . . . . . 12 |- ( ph -> ( A L R ) (⟂G ` G ) D )
69	68	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( A L R ) (⟂G ` G ) D )
70	69, 61	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( A L R ) (⟂G ` G ) ( R L t ) )
71	1, 2, 3, 4, 36, 45, 40, 66, 43, 70	perprag 25618	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> <" A R t "> e. (∟G ` G ) )
72		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. ( A I C ) )
73	1, 2, 3, 36, 45, 43, 38, 72	tgbtwncom 25383	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. ( C I A ) )
74	1, 2, 3, 4, 34, 36, 38, 40, 43, 45, 63, 71, 73	ragflat2 25598	. . . . . . . 8 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R = t )
75	33, 74	pm2.61dane 2881	. . . . . . 7 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R = t )
76	10	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> U e. P )
77	22	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> V e. P )
78	17	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> A ( K ` R ) U )
79	25	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> V ( K ` S ) C )
80	48	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> ( K ` R ) = ( K ` S ) )
81	80	breqd 4664	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> ( V ( K ` R ) C <-> V ( K ` S ) C ) )
82	79, 81	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> V ( K ` R ) C )
83	1, 3, 7, 77, 37, 39, 35, 82	hlcomd 25499	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> C ( K ` R ) V )
84		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. ( A I C ) )
85	75, 84	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( A I C ) )
86	1, 2, 3, 35, 44, 39, 37, 85	tgbtwncom 25383	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( C I A ) )
87	1, 3, 7, 37, 77, 44, 35, 39, 83, 86	btwnhl 25509	. . . . . . . . 9 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( V I A ) )
88	1, 2, 3, 35, 77, 39, 44, 87	tgbtwncom 25383	. . . . . . . 8 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( A I V ) )
89	1, 3, 7, 44, 76, 77, 35, 39, 78, 88	btwnhl 25509	. . . . . . 7 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( U I V ) )
90	75, 89	eqeltrrd 2702	. . . . . 6 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. ( U I V ) )
91		rspe 3003	. . . . . 6 |- ( ( t e. D /\ t e. ( U I V ) ) -> E. t e. D t e. ( U I V ) )
92	32, 90, 91	syl2anc 693	. . . . 5 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> E. t e. D t e. ( U I V ) )
93	19, 31, 92	jca31 557	. . . 4 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> ( ( -. U e. D /\ -. V e. D ) /\ E. t e. D t e. ( U I V ) ) )
94		hpg.o	. . . . . 6 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
95	1, 2, 3, 94, 10, 22	islnopp 25631	. . . . 5 |- ( ph -> ( U O V <-> ( ( -. U e. D /\ -. V e. D ) /\ E. t e. D t e. ( U I V ) ) ) )
96	95	ad3antrrr 766	. . . 4 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> ( U O V <-> ( ( -. U e. D /\ -. V e. D ) /\ E. t e. D t e. ( U I V ) ) ) )
97	93, 96	mpbird 247	. . 3 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> U O V )
98		opphllem5.o	. . . . . 6 |- ( ph -> A O C )
99	1, 2, 3, 94, 9, 21	islnopp 25631	. . . . . 6 |- ( ph -> ( A O C <-> ( ( -. A e. D /\ -. C e. D ) /\ E. t e. D t e. ( A I C ) ) ) )
100	98, 99	mpbid 222	. . . . 5 |- ( ph -> ( ( -. A e. D /\ -. C e. D ) /\ E. t e. D t e. ( A I C ) ) )
101	100	simprd 479	. . . 4 |- ( ph -> E. t e. D t e. ( A I C ) )
102	101	adantr 481	. . 3 |- ( ( ph /\ R = S ) -> E. t e. D t e. ( A I C ) )
103	97, 102	r19.29a 3078	. 2 |- ( ( ph /\ R = S ) -> U O V )
104	6	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> D e. ran L )
105	5	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> G e. TarskiG )
106		eqid 2622	. . . . 5 |- ( (pInvG ` G ) ` m ) = ( (pInvG ` G ) ` m )
107	9	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> A e. P )
108	21	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> C e. P )
109	8	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> R e. D )
110	20	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> S e. D )
111		simpllr 799	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> m e. P )
112	98	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> A O C )
113	11	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> D (⟂G ` G ) ( A L R ) )
114	23	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> D (⟂G ` G ) ( C L S ) )
115		simpr 477	. . . . . 6 |- ( ( ph /\ R =/= S ) -> R =/= S )
116	115	ad3antrrr 766	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> R =/= S )
117		simpr 477	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> ( S .- C ) (≤G ` G ) ( R .- A ) )
118	10	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> U e. P )
119		simplr 792	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> S = ( ( (pInvG ` G ) ` m ) ` R ) )
120	119	eqcomd 2628	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> ( ( (pInvG ` G ) ` m ) ` R ) = S )
121	22	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> V e. P )
122	13	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> U ( K ` R ) A )
123	25	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> V ( K ` S ) C )
124	1, 2, 3, 94, 4, 104, 105, 7, 106, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 120, 121, 122, 123	opphllem4 25642	. . . 4 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> U O V )
125	6	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> D e. ran L )
126	5	adantr 481	. . . . . 6 |- ( ( ph /\ R =/= S ) -> G e. TarskiG )
127	126	ad3antrrr 766	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> G e. TarskiG )
128	22	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> V e. P )
129	10	ad4antr 768	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> U e. P )
130	21	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> C e. P )
131	9	adantr 481	. . . . . . 7 |- ( ( ph /\ R =/= S ) -> A e. P )
132	131	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> A e. P )
133	20	adantr 481	. . . . . . 7 |- ( ( ph /\ R =/= S ) -> S e. D )
134	133	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> S e. D )
135	8	adantr 481	. . . . . . 7 |- ( ( ph /\ R =/= S ) -> R e. D )
136	135	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> R e. D )
137		simpllr 799	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> m e. P )
138	98	ad4antr 768	. . . . . . 7 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> A O C )
139	1, 2, 3, 94, 4, 125, 127, 132, 130, 138	oppcom 25636	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> C O A )
140	23	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> D (⟂G ` G ) ( C L S ) )
141	11	adantr 481	. . . . . . 7 |- ( ( ph /\ R =/= S ) -> D (⟂G ` G ) ( A L R ) )
142	141	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> D (⟂G ` G ) ( A L R ) )
143	115	necomd 2849	. . . . . . 7 |- ( ( ph /\ R =/= S ) -> S =/= R )
144	143	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> S =/= R )
145		simpr 477	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( R .- A ) (≤G ` G ) ( S .- C ) )
146	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ R =/= S ) -> R e. P )
147	146	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> R e. P )
148		simplr 792	. . . . . . . 8 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> S = ( ( (pInvG ` G ) ` m ) ` R ) )
149	148	eqcomd 2628	. . . . . . 7 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( ( (pInvG ` G ) ` m ) ` R ) = S )
150	1, 2, 3, 4, 34, 127, 137, 106, 147, 149	mircom 25558	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( ( (pInvG ` G ) ` m ) ` S ) = R )
151	25	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> V ( K ` S ) C )
152	13	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> U ( K ` R ) A )
153	1, 2, 3, 94, 4, 125, 127, 7, 106, 130, 132, 134, 136, 137, 139, 140, 142, 144, 145, 128, 150, 129, 151, 152	opphllem4 25642	. . . . 5 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> V O U )
154	1, 2, 3, 94, 4, 125, 127, 128, 129, 153	oppcom 25636	. . . 4 |- ( ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> U O V )
155		eqid 2622	. . . . . 6 |- (≤G ` G ) = (≤G ` G )
156	1, 2, 3, 155, 5, 24, 21, 12, 9	legtrid 25486	. . . . 5 |- ( ph -> ( ( S .- C ) (≤G ` G ) ( R .- A ) \/ ( R .- A ) (≤G ` G ) ( S .- C ) ) )
157	156	ad3antrrr 766	. . . 4 |- ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) -> ( ( S .- C ) (≤G ` G ) ( R .- A ) \/ ( R .- A ) (≤G ` G ) ( S .- C ) ) )
158	124, 154, 157	mpjaodan 827	. . 3 |- ( ( ( ( ph /\ R =/= S ) /\ m e. P ) /\ S = ( ( (pInvG ` G ) ` m ) ` R ) ) -> U O V )
159	24	adantr 481	. . . 4 |- ( ( ph /\ R =/= S ) -> S e. P )
160	1, 2, 3, 94, 4, 6, 5, 9, 21, 98	opptgdim2 25637	. . . . 5 |- ( ph -> GDimTarskiG≥ 2 )
161	160	adantr 481	. . . 4 |- ( ( ph /\ R =/= S ) -> GDimTarskiG≥ 2 )
162	1, 2, 3, 4, 126, 34, 146, 159, 161	midex 25629	. . 3 |- ( ( ph /\ R =/= S ) -> E. m e. P S = ( ( (pInvG ` G ) ` m ) ` R ) )
163	158, 162	r19.29a 3078	. 2 |- ( ( ph /\ R =/= S ) -> U O V )
164	103, 163	pm2.61dane 2881	1 |- ( ph -> U O V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ 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   2c2 11070   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  ≤Gcleg 25477  hlGchlg 25495  pInvGcmir 25547  ⟂Gcperpg 25590

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

This theorem is referenced by:  opphl  25646