Theorem colperpexlem3 24830

Description: Lemma for colperpex 24831. 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 2462	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
6		colperpex.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	ad2antrr 737	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> G e. TarskiG )
8		eqid 2462	. . . 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 24731	. . . . . . 7 |- ( ph -> ( A L B ) e. ran L )
13	12	ad2antrr 737	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A L B ) e. ran L )
14		simplr 767	. . . . . 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 24650	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> x e. P )
16		eqid 2462	. . . . 5 |- ( (pInvG ` G ) ` x ) = ( (pInvG ` G ) ` x )
17		colperpex.3	. . . . . 6 |- ( ph -> C e. P )
18	17	ad2antrr 737	. . . . 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 24762	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( ( (pInvG ` G ) ` x ) ` C ) e. P )
20	9	ad2antrr 737	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A e. P )
21		eqid 2462	. . . . 5 |- ( (pInvG ` G ) ` A ) = ( (pInvG ` G ) ` A )
22	1, 2, 3, 4, 5, 7, 20, 21, 18	mircl 24762	. . . 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 24758	. . . . 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 737	. . . . . . 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 737	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> -. C e. ( A L B ) )
27		nelne2 2733	. . . . . . . . . 10 |- ( ( x e. ( A L B ) /\ -. C e. ( A L B ) ) -> x =/= C )
28	14, 26, 27	syl2anc 671	. . . . . . . . 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 24731	. . . . . . . 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 24736	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) = ( C L x ) )
31		simpr 467	. . . . . . . . 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 4439	. . . . . . . 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 24814	. . . . . . 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 24824	. . . . . 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 24798	. . . . . 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 215	. . . . 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 2497	. . . 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 24793	. . 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 737	. . . . . . . 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 737	. . . . . . . 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 737	. . . . . . . 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 737	. . . . . . . 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 737	. . . . . . . 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 737	. . . . . . . 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 767	. . . . . . . 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 24759	. . . . . . . 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 24759	. . . . . . . 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 24759	. . . . . . . . 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 467	. . . . . . . . . . 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 2468	. . . . . . . . . 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 6335	. . . . . . . . 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 2542	. . . . . . . 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 24599	. . . . . . 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 741	. . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . 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 6336	. . . . . . . . . . . . . . . . 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 2542	. . . . . . . . . . . . . . . 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 24570	. . . . . . . . . . . . . . 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 2468	. . . . . . . . . . . . . 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 6335	. . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . 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 5896	. . . . . . . . . . . . . . . . . 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 5894	. . . . . . . . . . . . . . . . 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 2499	. . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . 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 24767	. . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . 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 6335	. . . . . . . . . . . . . . . . . 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 2543	. . . . . . . . . . . . . . . . 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 24570	. . . . . . . . . . . . . . . 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 2468	. . . . . . . . . . . . . . 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 6338	. . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . 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 24764	. . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . 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 24829	. . . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . 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 2703	. . . . . . . . . . . . . . 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 24736	. . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . . 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 24769	. . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . . 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 2499	. . . . . . . . . . . . . . . . . . 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 24766	. . . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . . 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 2640	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 2642	. . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . 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 24721	. . . . . . . . . . . . . . . 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 24733	. . . . . . . . . . . . . . 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 2691	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> C =/= x )
102	101	ad5antr 745	. . . . . . . . . . . . . . . 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 24736	. . . . . . . . . . . . . . 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 24736	. . . . . . . . . . . . . . 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 2506	. . . . . . . . . . . . . 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 2506	. . . . . . . . . . . . 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 2498	. . . . . . . . . . . 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 745	. . . . . . . . . . . 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 4439	. . . . . . . . . . 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 741	. . . . . . . . . . . 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 741	. . . . . . . . . . . . 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 741	. . . . . . . . . . . . 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 741	. . . . . . . . . . . . 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 24731	. . . . . . . . . . . 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 745	. . . . . . . . . . . 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 24734	. . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . 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 24734	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A e. ( A L B ) )
119	118	ad5antr 745	. . . . . . . . . . . . 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 3630	. . . . . . . . . . . 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 24735	. . . . . . . . . . . 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 745	. . . . . . . . . . . 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 2691	. . . . . . . . . . . 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 467	. . . . . . . . . . . 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 741	. . . . . . . . . . . . 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 24828	. . . . . . . . . . . . . 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 741	. . . . . . . . . . . . 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 24799	. . . . . . . . . . . 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 24818	. . . . . . . . . . 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 2723	. . . . . . . . . 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 745	. . . . . . . . . . . . 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 2540	. . . . . . . . . . . 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 398	. . . . . . . . . . 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 745	. . . . . . . . . . . . 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 745	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 2691	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 24720	. . . . . . . . . . . . 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 24732	. . . . . . . . . . . 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 398	. . . . . . . . . . 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 2723	. . . . . . . . . 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 737	. . . . . . . . . . 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 737	. . . . . . . . . . 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 767	. . . . . . . . . . 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 737	. . . . . . . . . . 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 769	. . . . . . . . . . 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 24588	. . . . . . . . . 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 542	. . . . . . . . 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 440	. . . . . . . 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 2874	. . . . . . 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 2957	. . . . . 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 201	. . . . 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 440	. . . 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 2874	. . 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 24819	. 2 |- ( ph -> E. x e. ( A L B ) ( C L x ) (⟂G ` G ) ( A L B ) )
159	157, 158	r19.29a 2944	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 374   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   E.wrex 2750   class class class wbr 4418   ran crn 4857   ` cfv 5605  (class class class)co 6320   <"cs3 12981   Basecbs 15176   distcds 15254  TarskiGcstrkg 24534  Itvcitv 24540  LineGclng 24541  pInvGcmir 24753  ∟Gcrag 24794  ⟂Gcperpg 24796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-pm 7506  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-card 8404  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-fzo 11953  df-hash 12554  df-word 12703  df-concat 12705  df-s1 12706  df-s2 12987  df-s3 12988  df-trkgc 24552  df-trkgb 24553  df-trkgcb 24554  df-trkg 24557  df-cgrg 24612  df-leg 24684  df-mir 24754  df-rag 24795  df-perpg 24797

This theorem is referenced by:  colperpex  24831