Theorem colperpexlem3 25624

Description: Lemma for colperpex 25625. Case 1 of theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)

Hypotheses

Ref	Expression
colperpex.p	|- P = ( Base ` G )
colperpex.d	|- .- = ( dist ` G )
colperpex.i	|- I = (Itv ` G )
colperpex.l	|- L = (LineG ` G )
colperpex.g	|- ( ph -> G e. TarskiG )
colperpex.1	|- ( ph -> A e. P )
colperpex.2	|- ( ph -> B e. P )
colperpex.3	|- ( ph -> C e. P )
colperpex.4	|- ( ph -> A =/= B )
colperpexlem3.1	|- ( ph -> -. C e. ( A L B ) )

Assertion

Ref	Expression
colperpexlem3	|- ( ph -> E. p e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )

Distinct variable groups:   .- , p, t   A, p, t   B, p, t   C, p, t   G, p, t   I, p, t   L, p, t   P, p, t   ph, p, t

Proof of Theorem colperpexlem3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		colperpex.p	. . . 4 |- P = ( Base ` G )
2		colperpex.d	. . . 4 |- .- = ( dist ` G )
3		colperpex.i	. . . 4 |- I = (Itv ` G )
4		colperpex.l	. . . 4 |- L = (LineG ` G )
5		eqid 2622	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
6		colperpex.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	ad2antrr 762	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> G e. TarskiG )
8		eqid 2622	. . . 4 |- ( (pInvG ` G ) ` p ) = ( (pInvG ` G ) ` p )
9		colperpex.1	. . . . . . . 8 |- ( ph -> A e. P )
10		colperpex.2	. . . . . . . 8 |- ( ph -> B e. P )
11		colperpex.4	. . . . . . . 8 |- ( ph -> A =/= B )
12	1, 3, 4, 6, 9, 10, 11	tgelrnln 25525	. . . . . . 7 |- ( ph -> ( A L B ) e. ran L )
13	12	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A L B ) e. ran L )
14		simplr 792	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> x e. ( A L B ) )
15	1, 4, 3, 7, 13, 14	tglnpt 25444	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> x e. P )
16		eqid 2622	. . . . 5 |- ( (pInvG ` G ) ` x ) = ( (pInvG ` G ) ` x )
17		colperpex.3	. . . . . 6 |- ( ph -> C e. P )
18	17	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> C e. P )
19	1, 2, 3, 4, 5, 7, 15, 16, 18	mircl 25556	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( ( (pInvG ` G ) ` x ) ` C ) e. P )
20	9	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A e. P )
21		eqid 2622	. . . . 5 |- ( (pInvG ` G ) ` A ) = ( (pInvG ` G ) ` A )
22	1, 2, 3, 4, 5, 7, 20, 21, 18	mircl 25556	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( ( (pInvG ` G ) ` A ) ` C ) e. P )
23	1, 2, 3, 4, 5, 7, 20, 21, 18	mircgr 25552	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A .- ( ( (pInvG ` G ) ` A ) ` C ) ) = ( A .- C ) )
24	10	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> B e. P )
25		colperpexlem3.1	. . . . . . . . . . 11 |- ( ph -> -. C e. ( A L B ) )
26	25	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> -. C e. ( A L B ) )
27		nelne2 2891	. . . . . . . . . 10 |- ( ( x e. ( A L B ) /\ -. C e. ( A L B ) ) -> x =/= C )
28	14, 26, 27	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> x =/= C )
29	1, 3, 4, 7, 15, 18, 28	tgelrnln 25525	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) e. ran L )
30	1, 3, 4, 7, 15, 18, 28	tglinecom 25530	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) = ( C L x ) )
31		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( C L x ) (⟂G ` G ) ( A L B ) )
32	30, 31	eqbrtrd 4675	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) (⟂G ` G ) ( A L B ) )
33	1, 2, 3, 4, 7, 29, 13, 32	perpcom 25608	. . . . . . 7 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A L B ) (⟂G ` G ) ( x L C ) )
34	1, 2, 3, 4, 7, 20, 24, 14, 18, 33	perprag 25618	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> <" A x C "> e. (∟G ` G ) )
35	1, 2, 3, 4, 5, 7, 20, 15, 18	israg 25592	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( <" A x C "> e. (∟G ` G ) <-> ( A .- C ) = ( A .- ( ( (pInvG ` G ) ` x ) ` C ) ) ) )
36	34, 35	mpbid 222	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A .- C ) = ( A .- ( ( (pInvG ` G ) ` x ) ` C ) ) )
37	23, 36	eqtr2d 2657	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A .- ( ( (pInvG ` G ) ` x ) ` C ) ) = ( A .- ( ( (pInvG ` G ) ` A ) ` C ) ) )
38	1, 2, 3, 4, 5, 7, 8, 19, 22, 20, 37	midexlem 25587	. . 3 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> E. p e. P ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) )
39	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> G e. TarskiG )
40	22	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( (pInvG ` G ) ` A ) ` C ) e. P )
41	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> A e. P )
42	18	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> C e. P )
43	19	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( (pInvG ` G ) ` x ) ` C ) e. P )
44	15	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> x e. P )
45		simplr 792	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> p e. P )
46	1, 2, 3, 4, 5, 39, 41, 21, 42	mirbtwn 25553	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> A e. ( ( ( (pInvG ` G ) ` A ) ` C ) I C ) )
47	1, 2, 3, 4, 5, 39, 44, 16, 42	mirbtwn 25553	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> x e. ( ( ( (pInvG ` G ) ` x ) ` C ) I C ) )
48	1, 2, 3, 4, 5, 39, 45, 8, 43	mirbtwn 25553	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> p e. ( ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) I ( ( (pInvG ` G ) ` x ) ` C ) ) )
49		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) )
50	49	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) = ( ( (pInvG ` G ) ` A ) ` C ) )
51	50	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) I ( ( (pInvG ` G ) ` x ) ` C ) ) = ( ( ( (pInvG ` G ) ` A ) ` C ) I ( ( (pInvG ` G ) ` x ) ` C ) ) )
52	48, 51	eleqtrd 2703	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> p e. ( ( ( (pInvG ` G ) ` A ) ` C ) I ( ( (pInvG ` G ) ` x ) ` C ) ) )
53	1, 2, 3, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52	tgtrisegint 25394	. . . . . . 7 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> E. t e. P ( t e. ( p I C ) /\ t e. ( A I x ) ) )
54	39	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> G e. TarskiG )
55	41	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> A e. P )
56		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t e. P )
57		simplrr 801	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t e. ( A I x ) )
58		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> x = A )
59	58	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( A I x ) = ( A I A ) )
60	57, 59	eleqtrd 2703	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t e. ( A I A ) )
61	1, 2, 3, 54, 55, 56, 60	axtgbtwnid 25365	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> A = t )
62	61	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t = A )
63	62	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( t L p ) = ( A L p ) )
64	50	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) = ( ( (pInvG ` G ) ` A ) ` C ) )
65	58	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( (pInvG ` G ) ` x ) = ( (pInvG ` G ) ` A ) )
66	65	fveq1d 6193	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( (pInvG ` G ) ` x ) ` C ) = ( ( (pInvG ` G ) ` A ) ` C ) )
67	64, 66	eqtr4d 2659	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) = ( ( (pInvG ` G ) ` x ) ` C ) )
68	45	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> p e. P )
69	43	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( (pInvG ` G ) ` x ) ` C ) e. P )
70	1, 2, 3, 4, 5, 54, 68, 8, 69	mirinv 25561	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) = ( ( (pInvG ` G ) ` x ) ` C ) <-> p = ( ( (pInvG ` G ) ` x ) ` C ) ) )
71	67, 70	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> p = ( ( (pInvG ` G ) ` x ) ` C ) )
72	44	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> x e. P )
73	58	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( x I x ) = ( A I x ) )
74	57, 73	eleqtrrd 2704	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t e. ( x I x ) )
75	1, 2, 3, 54, 72, 56, 74	axtgbtwnid 25365	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> x = t )
76	75	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t = x )
77	71, 76	oveq12d 6668	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( p L t ) = ( ( ( (pInvG ` G ) ` x ) ` C ) L x ) )
78	34	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> <" A x C "> e. (∟G ` G ) )
79	1, 2, 3, 4, 5, 39, 45, 8, 43, 50	mircom 25558	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` A ) ` C ) ) = ( ( (pInvG ` G ) ` x ) ` C ) )
80	28	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> x =/= C )
81	1, 2, 3, 4, 39, 5, 21, 16, 8, 41, 44, 42, 45, 78, 79, 80	colperpexlem2 25623	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> A =/= p )
82	81	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> A =/= p )
83	62, 82	eqnetrd 2861	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t =/= p )
84	1, 3, 4, 54, 56, 68, 83	tglinecom 25530	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( t L p ) = ( p L t ) )
85	42	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> C e. P )
86	80	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> x =/= C )
87	54	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> G e. TarskiG )
88	72	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> x e. P )
89	85	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> C e. P )
90	1, 2, 3, 4, 5, 87, 88, 16	mircinv 25563	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> ( ( (pInvG ` G ) ` x ) ` x ) = x )
91		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> ( ( (pInvG ` G ) ` x ) ` C ) = x )
92	90, 91	eqtr4d 2659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> ( ( (pInvG ` G ) ` x ) ` x ) = ( ( (pInvG ` G ) ` x ) ` C ) )
93	1, 2, 3, 4, 5, 87, 88, 16, 88, 89, 92	mireq 25560	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> x = C )
94	86	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> x =/= C )
95	94	neneqd 2799	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) /\ ( ( (pInvG ` G ) ` x ) ` C ) = x ) -> -. x = C )
96	93, 95	pm2.65da 600	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> -. ( ( (pInvG ` G ) ` x ) ` C ) = x )
97	96	neqned 2801	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( (pInvG ` G ) ` x ) ` C ) =/= x )
98	47	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> x e. ( ( ( (pInvG ` G ) ` x ) ` C ) I C ) )
99	1, 3, 4, 54, 72, 85, 69, 86, 98	btwnlng2 25515	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( (pInvG ` G ) ` x ) ` C ) e. ( x L C ) )
100	1, 3, 4, 54, 72, 85, 86, 69, 97, 99	tglineelsb2 25527	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( x L C ) = ( x L ( ( (pInvG ` G ) ` x ) ` C ) ) )
101	28	necomd 2849	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> C =/= x )
102	101	ad5antr 770	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> C =/= x )
103	1, 3, 4, 54, 85, 72, 102	tglinecom 25530	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( C L x ) = ( x L C ) )
104	1, 3, 4, 54, 69, 72, 97	tglinecom 25530	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( ( ( (pInvG ` G ) ` x ) ` C ) L x ) = ( x L ( ( (pInvG ` G ) ` x ) ` C ) ) )
105	100, 103, 104	3eqtr4d 2666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( C L x ) = ( ( ( (pInvG ` G ) ` x ) ` C ) L x ) )
106	77, 84, 105	3eqtr4d 2666	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( t L p ) = ( C L x ) )
107	63, 106	eqtr3d 2658	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( A L p ) = ( C L x ) )
108	31	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( C L x ) (⟂G ` G ) ( A L B ) )
109	107, 108	eqbrtrd 4675	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( A L p ) (⟂G ` G ) ( A L B ) )
110	39	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> G e. TarskiG )
111	41	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A e. P )
112	45	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> p e. P )
113	81	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A =/= p )
114	1, 3, 4, 110, 111, 112, 113	tgelrnln 25525	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> ( A L p ) e. ran L )
115	13	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> ( A L B ) e. ran L )
116	1, 3, 4, 110, 111, 112, 113	tglinerflx1 25528	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A e. ( A L p ) )
117	11	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A =/= B )
118	1, 3, 4, 7, 20, 24, 117	tglinerflx1 25528	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A e. ( A L B ) )
119	118	ad5antr 770	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A e. ( A L B ) )
120	116, 119	elind 3798	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A e. ( ( A L p ) i^i ( A L B ) ) )
121	1, 3, 4, 110, 111, 112, 113	tglinerflx2 25529	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> p e. ( A L p ) )
122	14	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> x e. ( A L B ) )
123	113	necomd 2849	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> p =/= A )
124		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> x =/= A )
125	44	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> x e. P )
126	1, 2, 3, 4, 39, 5, 21, 16, 8, 41, 44, 42, 45, 78, 79	colperpexlem1 25622	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> <" x A p "> e. (∟G ` G ) )
127	126	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> <" x A p "> e. (∟G ` G ) )
128	1, 2, 3, 4, 5, 110, 125, 111, 112, 127	ragcom 25593	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> <" p A x "> e. (∟G ` G ) )
129	1, 2, 3, 4, 110, 114, 115, 120, 121, 122, 123, 124, 128	ragperp 25612	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> ( A L p ) (⟂G ` G ) ( A L B ) )
130	109, 129	pm2.61dane 2881	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> ( A L p ) (⟂G ` G ) ( A L B ) )
131	118	ad5antr 770	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> A e. ( A L B ) )
132	62, 131	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> t e. ( A L B ) )
133	132	orcd 407	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x = A ) -> ( t e. ( A L B ) \/ A = B ) )
134	24	ad5antr 770	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> B e. P )
135	117	ad5antr 770	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A =/= B )
136		simpllr 799	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> t e. P )
137	124	necomd 2849	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> A =/= x )
138		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> t e. ( A I x ) )
139	1, 3, 4, 110, 111, 125, 136, 137, 138	btwnlng1 25514	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> t e. ( A L x ) )
140	1, 3, 4, 110, 111, 134, 135, 125, 124, 122, 136, 139	tglineeltr 25526	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> t e. ( A L B ) )
141	140	orcd 407	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) /\ x =/= A ) -> ( t e. ( A L B ) \/ A = B ) )
142	133, 141	pm2.61dane 2881	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> ( t e. ( A L B ) \/ A = B ) )
143	39	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> G e. TarskiG )
144	45	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> p e. P )
145		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> t e. P )
146	42	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> C e. P )
147		simprl 794	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> t e. ( p I C ) )
148	1, 2, 3, 143, 144, 145, 146, 147	tgbtwncom 25383	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> t e. ( C I p ) )
149	130, 142, 148	jca32 558	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) /\ ( t e. ( p I C ) /\ t e. ( A I x ) ) ) -> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )
150	149	ex 450	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) /\ t e. P ) -> ( ( t e. ( p I C ) /\ t e. ( A I x ) ) -> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) ) )
151	150	reximdva 3017	. . . . . . 7 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( E. t e. P ( t e. ( p I C ) /\ t e. ( A I x ) ) -> E. t e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) ) )
152	53, 151	mpd 15	. . . . . 6 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> E. t e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )
153		r19.42v 3092	. . . . . 6 |- ( E. t e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) <-> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )
154	152, 153	sylib 208	. . . . 5 |- ( ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) ) -> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )
155	154	ex 450	. . . 4 |- ( ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) -> ( ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) -> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) ) )
156	155	reximdva 3017	. . 3 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( E. p e. P ( ( (pInvG ` G ) ` A ) ` C ) = ( ( (pInvG ` G ) ` p ) ` ( ( (pInvG ` G ) ` x ) ` C ) ) -> E. p e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) ) )
157	38, 156	mpd 15	. 2 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> E. p e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )
158	1, 2, 3, 4, 6, 12, 17, 25	footex 25613	. 2 |- ( ph -> E. x e. ( A L B ) ( C L x ) (⟂G ` G ) ( A L B ) )
159	157, 158	r19.29a 3078	1 |- ( ph -> E. p e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( C I p ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂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-trkg 25352  df-cgrg 25406  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  colperpex  25625