Theorem colperpexlem3 23926

Description: Lemma for colperpex 23927. 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 2467	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
6		colperpex.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	ad2antrr 725	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> G e. TarskiG )
8		eqid 2467	. . . 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 23839	. . . . . . 7 |- ( ph -> ( A L B ) e. ran L )
13	12	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A L B ) e. ran L )
14		simplr 754	. . . . . 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 23779	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> x e. P )
16		eqid 2467	. . . . 5 |- ( (pInvG ` G ) ` x ) = ( (pInvG ` G ) ` x )
17		colperpex.3	. . . . . 6 |- ( ph -> C e. P )
18	17	ad2antrr 725	. . . . 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 23870	. . . 4 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( ( (pInvG ` G ) ` x ) ` C ) e. P )
20	9	ad2antrr 725	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A e. P )
21		eqid 2467	. . . . 5 |- ( (pInvG ` G ) ` A ) = ( (pInvG ` G ) ` A )
22	1, 2, 3, 4, 5, 7, 20, 21, 18	mircl 23870	. . . 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 23866	. . . . 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 725	. . . . . . 7 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> B e. P )
25		colperpexlem3.1	. . . . . . . . . . . 12 |- ( ph -> -. C e. ( A L B ) )
26	25	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> -. C e. ( A L B ) )
27		nelne2 2797	. . . . . . . . . . 11 |- ( ( x e. ( A L B ) /\ -. C e. ( A L B ) ) -> x =/= C )
28	14, 26, 27	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> x =/= C )
29	1, 3, 4, 7, 15, 18, 28	tglinecom 23844	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) = ( C L x ) )
30	28	necomd 2738	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> C =/= x )
31	1, 3, 4, 7, 18, 15, 30	tgelrnln 23839	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( C L x ) e. ran L )
32	29, 31	eqeltrd 2555	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) e. ran L )
33		simpr 461	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( C L x ) (⟂G ` G ) ( A L B ) )
34	29, 33	eqbrtrd 4472	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( x L C ) (⟂G ` G ) ( A L B ) )
35	1, 2, 3, 4, 7, 32, 13, 34	perpcom 23913	. . . . . . 7 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A L B ) (⟂G ` G ) ( x L C ) )
36	1, 2, 3, 4, 7, 20, 24, 14, 18, 35	perprag 23920	. . . . . 6 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> <" A x C "> e. (∟G ` G ) )
37	1, 2, 3, 4, 5, 7, 20, 15, 18	israg 23897	. . . . . 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 ) ) ) )
38	36, 37	mpbid 210	. . . . 5 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> ( A .- C ) = ( A .- ( ( (pInvG ` G ) ` x ) ` C ) ) )
39	23, 38	eqtr2d 2509	. . . 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 ) ) )
40	1, 2, 3, 4, 5, 7, 8, 19, 22, 20, 39	midexlem 23892	. . 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 ) ) )
41	7	ad2antrr 725	. . . . . . . 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 )
42	22	ad2antrr 725	. . . . . . . 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 )
43	20	ad2antrr 725	. . . . . . . 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 )
44	18	ad2antrr 725	. . . . . . . 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 )
45	19	ad2antrr 725	. . . . . . . 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 )
46	15	ad2antrr 725	. . . . . . . 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 )
47		simplr 754	. . . . . . . 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 )
48	1, 2, 3, 4, 5, 41, 43, 21, 44	mirbtwn 23867	. . . . . . . 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 ) )
49	1, 2, 3, 4, 5, 41, 46, 16, 44	mirbtwn 23867	. . . . . . . 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 ) )
50	1, 2, 3, 4, 5, 41, 47, 8, 45	mirbtwn 23867	. . . . . . . . 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 ) ) )
51		simpr 461	. . . . . . . . . . 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 ) ) )
52	51	eqcomd 2475	. . . . . . . . . 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 ) )
53	52	oveq1d 6309	. . . . . . . . 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 ) ) )
54	50, 53	eleqtrd 2557	. . . . . . . 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 ) ) )
55	1, 2, 3, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54	tgtrisegint 23733	. . . . . . 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 ) ) )
56	41	ad3antrrr 729	. . . . . . . . . . . . . . . 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 )
57	43	ad3antrrr 729	. . . . . . . . . . . . . . . 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 )
58		simpllr 758	. . . . . . . . . . . . . . . 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 )
59		simplrr 760	. . . . . . . . . . . . . . . . 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 ) )
60		simpr 461	. . . . . . . . . . . . . . . . . 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 )
61	60	oveq2d 6310	. . . . . . . . . . . . . . . . 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 ) )
62	59, 61	eleqtrd 2557	. . . . . . . . . . . . . . . 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 ) )
63	1, 2, 3, 56, 57, 58, 62	axtgbtwnid 23706	. . . . . . . . . . . . . . 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 )
64	63	eqcomd 2475	. . . . . . . . . . . . . 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 )
65	64	oveq1d 6309	. . . . . . . . . . . . 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 ) )
66	52	ad3antrrr 729	. . . . . . . . . . . . . . . . 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 ) )
67	60	fveq2d 5875	. . . . . . . . . . . . . . . . . 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 ) )
68	67	fveq1d 5873	. . . . . . . . . . . . . . . . 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 ) )
69	66, 68	eqtr4d 2511	. . . . . . . . . . . . . . . 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 ) )
70	47	ad3antrrr 729	. . . . . . . . . . . . . . . . 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 )
71	45	ad3antrrr 729	. . . . . . . . . . . . . . . . 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 )
72	1, 2, 3, 4, 5, 56, 70, 8, 71	mirinv 23875	. . . . . . . . . . . . . . . 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 ) ) )
73	69, 72	mpbid 210	. . . . . . . . . . . . . . 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 ) )
74	46	ad3antrrr 729	. . . . . . . . . . . . . . . . 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 )
75	60	oveq1d 6309	. . . . . . . . . . . . . . . . . 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 ) )
76	59, 75	eleqtrrd 2558	. . . . . . . . . . . . . . . . 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 ) )
77	1, 2, 3, 56, 74, 58, 76	axtgbtwnid 23706	. . . . . . . . . . . . . . . 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 )
78	77	eqcomd 2475	. . . . . . . . . . . . . . 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 )
79	73, 78	oveq12d 6312	. . . . . . . . . . . . . 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 ) )
80	36	ad2antrr 725	. . . . . . . . . . . . . . . . . 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 ) )
81	1, 2, 3, 4, 5, 41, 47, 8, 45, 52	mircom 23872	. . . . . . . . . . . . . . . . . 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 ) )
82	28	ad2antrr 725	. . . . . . . . . . . . . . . . . 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 )
83	1, 2, 3, 4, 41, 5, 21, 16, 8, 43, 46, 44, 47, 80, 81, 82	colperpexlem2 23925	. . . . . . . . . . . . . . . . 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 )
84	83	ad3antrrr 729	. . . . . . . . . . . . . . . 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 )
85	64, 84	eqnetrd 2760	. . . . . . . . . . . . . . 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 )
86	1, 3, 4, 56, 58, 70, 85	tglinecom 23844	. . . . . . . . . . . . . 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 ) )
87	44	ad3antrrr 729	. . . . . . . . . . . . . . . 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 )
88	82	ad3antrrr 729	. . . . . . . . . . . . . . . 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 )
89	56	adantr 465	. . . . . . . . . . . . . . . . . . 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 )
90	74	adantr 465	. . . . . . . . . . . . . . . . . . 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 )
91	87	adantr 465	. . . . . . . . . . . . . . . . . . 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 )
92	1, 2, 3, 4, 5, 89, 90, 16	mircinv 23876	. . . . . . . . . . . . . . . . . . . 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 )
93		simpr 461	. . . . . . . . . . . . . . . . . . . 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 )
94	92, 93	eqtr4d 2511	. . . . . . . . . . . . . . . . . . 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 ) )
95	1, 2, 3, 4, 5, 89, 90, 16, 90, 91, 94	mireq 23874	. . . . . . . . . . . . . . . . . 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	88	adantr 465	. . . . . . . . . . . . . . . . . . 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 )
97	96	neneqd 2669	. . . . . . . . . . . . . . . . . 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 )
98	95, 97	pm2.65da 576	. . . . . . . . . . . . . . . . 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 )
99	98	neqned 2670	. . . . . . . . . . . . . . . 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 )
100	49	ad3antrrr 729	. . . . . . . . . . . . . . . . 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 ) )
101	1, 3, 4, 56, 74, 87, 71, 88, 100	btwnlng2 23829	. . . . . . . . . . . . . . . 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 ) )
102	1, 3, 4, 56, 74, 87, 88, 71, 99, 101	tglineelsb2 23841	. . . . . . . . . . . . . . 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 ) ) )
103	30	ad5antr 733	. . . . . . . . . . . . . . . 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 )
104	1, 3, 4, 56, 87, 74, 103	tglinecom 23844	. . . . . . . . . . . . . . 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 ) )
105	1, 3, 4, 56, 71, 74, 99	tglinecom 23844	. . . . . . . . . . . . . . 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 ) ) )
106	102, 104, 105	3eqtr4d 2518	. . . . . . . . . . . . . 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 ) )
107	79, 86, 106	3eqtr4d 2518	. . . . . . . . . . . . 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 ) )
108	65, 107	eqtr3d 2510	. . . . . . . . . . . 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 ) )
109	33	ad5antr 733	. . . . . . . . . . . 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 ) )
110	108, 109	eqbrtrd 4472	. . . . . . . . . . 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 ) )
111	41	ad3antrrr 729	. . . . . . . . . . . 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 )
112	43	ad3antrrr 729	. . . . . . . . . . . . 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 )
113	47	ad3antrrr 729	. . . . . . . . . . . . 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 )
114	83	ad3antrrr 729	. . . . . . . . . . . . 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 )
115	1, 3, 4, 111, 112, 113, 114	tgelrnln 23839	. . . . . . . . . . . 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 )
116	13	ad5antr 733	. . . . . . . . . . . 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 )
117	1, 3, 4, 111, 112, 113, 114	tglinerflx1 23842	. . . . . . . . . . . . 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 ) )
118	11	ad2antrr 725	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A =/= B )
119	1, 3, 4, 7, 20, 24, 118	tglinerflx1 23842	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A L B ) ) /\ ( C L x ) (⟂G ` G ) ( A L B ) ) -> A e. ( A L B ) )
120	119	ad5antr 733	. . . . . . . . . . . . 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 ) )
121	117, 120	elind 3693	. . . . . . . . . . . 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 ) ) )
122	1, 3, 4, 111, 112, 113, 114	tglinerflx2 23843	. . . . . . . . . . . 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 ) )
123	14	ad5antr 733	. . . . . . . . . . . 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 ) )
124	114	necomd 2738	. . . . . . . . . . . 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 )
125		simpr 461	. . . . . . . . . . . 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 )
126	46	ad3antrrr 729	. . . . . . . . . . . . 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 )
127	1, 2, 3, 4, 41, 5, 21, 16, 8, 43, 46, 44, 47, 80, 81	colperpexlem1 23924	. . . . . . . . . . . . . 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 ) )
128	127	ad3antrrr 729	. . . . . . . . . . . . 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 ) )
129	1, 2, 3, 4, 5, 111, 126, 112, 113, 128	ragcom 23898	. . . . . . . . . . . 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 ) )
130	1, 2, 3, 4, 111, 115, 116, 121, 122, 123, 124, 125, 129	ragperp 23917	. . . . . . . . . . 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 ) )
131	110, 130	pm2.61dane 2785	. . . . . . . . . 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 ) )
132	119	ad5antr 733	. . . . . . . . . . . . 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 ) )
133	64, 132	eqeltrd 2555	. . . . . . . . . . . 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 ) )
134	133	orcd 392	. . . . . . . . . . 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 ) )
135	24	ad5antr 733	. . . . . . . . . . . . 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 )
136	118	ad5antr 733	. . . . . . . . . . . . 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 )
137		simpllr 758	. . . . . . . . . . . . 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 )
138	125	necomd 2738	. . . . . . . . . . . . . 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 )
139		simplrr 760	. . . . . . . . . . . . . 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 ) )
140	1, 3, 4, 111, 112, 126, 137, 138, 139	btwnlng1 23828	. . . . . . . . . . . . 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 ) )
141	1, 3, 4, 111, 112, 135, 136, 126, 125, 123, 137, 140	tglineeltr 23840	. . . . . . . . . . . 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 ) )
142	141	orcd 392	. . . . . . . . . . 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 ) )
143	134, 142	pm2.61dane 2785	. . . . . . . . . 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 ) )
144	41	ad2antrr 725	. . . . . . . . . . 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 )
145	47	ad2antrr 725	. . . . . . . . . . 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 )
146		simplr 754	. . . . . . . . . . 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 )
147	44	ad2antrr 725	. . . . . . . . . . 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 )
148		simprl 755	. . . . . . . . . . 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 ) )
149	1, 2, 3, 144, 145, 146, 147, 148	tgbtwncom 23722	. . . . . . . . . 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 ) )
150	131, 143, 149	jca32 535	. . . . . . . . 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 ) ) ) )
151	150	ex 434	. . . . . . . 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 ) ) ) ) )
152	151	reximdva 2942	. . . . . . 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 ) ) ) ) )
153	55, 152	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 ) ) ) )
154		r19.42v 3021	. . . . . 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 ) ) ) )
155	153, 154	sylib 196	. . . . 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 ) ) ) )
156	155	ex 434	. . . 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 ) ) ) ) )
157	156	reximdva 2942	. . 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 ) ) ) ) )
158	40, 157	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 ) ) ) )
159	1, 2, 3, 4, 6, 12, 17, 25	footex 23918	. 2 |- ( ph -> E. x e. ( A L B ) ( C L x ) (⟂G ` G ) ( A L B ) )
160	158, 159	r19.29a 3008	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 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2818   class class class wbr 4452   ran crn 5005   ` cfv 5593  (class class class)co 6294   <"cs3 12782   Basecbs 14502   distcds 14576  TarskiGcstrkg 23668  Itvcitv 23675  LineGclng 23676  pInvGcmir 23861  ∟Gcrag 23893  ⟂Gcperpg 23895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-hash 12384  df-word 12518  df-concat 12520  df-s1 12521  df-s2 12788  df-s3 12789  df-trkgc 23687  df-trkgb 23688  df-trkgcb 23689  df-trkg 23693  df-cgrg 23746  df-leg 23812  df-mir 23862  df-rag 23894  df-perpg 23896

This theorem is referenced by:  colperpex  23927