Theorem lmieu 24354

Description: Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)

Hypotheses

Ref	Expression
ismid.p	|- P = ( Base ` G )
ismid.d	|- .- = ( dist ` G )
ismid.i	|- I = (Itv ` G )
ismid.g	|- ( ph -> G e. TarskiG )
ismid.1	|- ( ph -> GDimTarskiG≥ 2 )
lmieu.l	|- L = (LineG ` G )
lmieu.1	|- ( ph -> D e. ran L )
lmieu.a	|- ( ph -> A e. P )

Assertion

Ref	Expression
lmieu	|- ( ph -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )

Distinct variable groups:   G, b   P, b   ph, b   A, b   D, b   L, b

Allowed substitution hints:   I( b)   .- ( b)

Proof of Theorem lmieu

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmieu.a	. . . 4 |- ( ph -> A e. P )
2	1	adantr 463	. . 3 |- ( ( ph /\ A e. D ) -> A e. P )
3		simpr 459	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> -. A = b )
4		eqidd 2455	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> ( A (midG ` G ) b ) = ( A (midG ` G ) b ) )
5		ismid.p	. . . . . . . . . . . . . . . 16 |- P = ( Base ` G )
6		ismid.d	. . . . . . . . . . . . . . . 16 |- .- = ( dist ` G )
7		ismid.i	. . . . . . . . . . . . . . . 16 |- I = (Itv ` G )
8		ismid.g	. . . . . . . . . . . . . . . . 17 |- ( ph -> G e. TarskiG )
9	8	ad4antr 729	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> G e. TarskiG )
10		ismid.1	. . . . . . . . . . . . . . . . 17 |- ( ph -> GDimTarskiG≥ 2 )
11	10	ad4antr 729	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> GDimTarskiG≥ 2 )
12	2	ad3antrrr 727	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> A e. P )
13		simpllr 758	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> b e. P )
14		eqid 2454	. . . . . . . . . . . . . . . 16 |- (pInvG ` G ) = (pInvG ` G )
15	5, 6, 7, 9, 11, 12, 13	midcl 24347	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> ( A (midG ` G ) b ) e. P )
16	5, 6, 7, 9, 11, 12, 13, 14, 15	ismidb 24348	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> ( b = ( ( (pInvG ` G ) ` ( A (midG ` G ) b ) ) ` A ) <-> ( A (midG ` G ) b ) = ( A (midG ` G ) b ) ) )
17	4, 16	mpbird 232	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> b = ( ( (pInvG ` G ) ` ( A (midG ` G ) b ) ) ` A ) )
18	17	adantr 463	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> b = ( ( (pInvG ` G ) ` ( A (midG ` G ) b ) ) ` A ) )
19		lmieu.l	. . . . . . . . . . . . . . . 16 |- L = (LineG ` G )
20	9	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> G e. TarskiG )
21		lmieu.1	. . . . . . . . . . . . . . . . . 18 |- ( ph -> D e. ran L )
22	21	ad4antr 729	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> D e. ran L )
23	22	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> D e. ran L )
24	12	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A e. P )
25	13	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> b e. P )
26	3	neqned 2657	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> A =/= b )
27	26	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A =/= b )
28	5, 7, 19, 20, 24, 25, 27	tgelrnln 24214	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( A L b ) e. ran L )
29		simpr 459	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> D =/= ( A L b ) )
30		simp-4r 766	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> A e. D )
31	30	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A e. D )
32	5, 7, 19, 20, 24, 25, 27	tglinerflx1 24217	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A e. ( A L b ) )
33	31, 32	elind 3674	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A e. ( D i^i ( A L b ) ) )
34		simpllr 758	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( A (midG ` G ) b ) e. D )
35	5, 6, 7, 9, 11, 12, 13	midbtwn 24349	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> ( A (midG ` G ) b ) e. ( A I b ) )
36	5, 7, 19, 9, 12, 13, 15, 26, 35	btwnlng1 24203	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> ( A (midG ` G ) b ) e. ( A L b ) )
37	36	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( A (midG ` G ) b ) e. ( A L b ) )
38	34, 37	elind 3674	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( A (midG ` G ) b ) e. ( D i^i ( A L b ) ) )
39	5, 7, 19, 20, 23, 28, 29, 33, 38	tglineineq 24227	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A = ( A (midG ` G ) b ) )
40	39	fveq2d 5852	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( (pInvG ` G ) ` A ) = ( (pInvG ` G ) ` ( A (midG ` G ) b ) ) )
41	40	fveq1d 5850	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( ( (pInvG ` G ) ` A ) ` A ) = ( ( (pInvG ` G ) ` ( A (midG ` G ) b ) ) ` A ) )
42		eqid 2454	. . . . . . . . . . . . . 14 |- ( (pInvG ` G ) ` A ) = ( (pInvG ` G ) ` A )
43	5, 6, 7, 19, 14, 20, 24, 42	mircinv 24252	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> ( ( (pInvG ` G ) ` A ) ` A ) = A )
44	18, 41, 43	3eqtr2rd 2502	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D =/= ( A L b ) ) -> A = b )
45	3, 44	mtand 657	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> -. D =/= ( A L b ) )
46	8	ad5antr 731	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D (⟂G ` G ) ( A L b ) ) -> G e. TarskiG )
47	21	ad5antr 731	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D (⟂G ` G ) ( A L b ) ) -> D e. ran L )
48		nne 2655	. . . . . . . . . . . . . . 15 |- ( -. D =/= ( A L b ) <-> D = ( A L b ) )
49	45, 48	sylib 196	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> D = ( A L b ) )
50	49	adantr 463	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D (⟂G ` G ) ( A L b ) ) -> D = ( A L b ) )
51	50, 47	eqeltrrd 2543	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D (⟂G ` G ) ( A L b ) ) -> ( A L b ) e. ran L )
52		simpr 459	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D (⟂G ` G ) ( A L b ) ) -> D (⟂G ` G ) ( A L b ) )
53	5, 6, 7, 19, 46, 47, 51, 52	perpneq 24295	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) /\ D (⟂G ` G ) ( A L b ) ) -> D =/= ( A L b ) )
54	45, 53	mtand 657	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> -. D (⟂G ` G ) ( A L b ) )
55	54	ex 432	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) -> ( -. A = b -> -. D (⟂G ` G ) ( A L b ) ) )
56	55	con4d 105	. . . . . . . 8 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) -> ( D (⟂G ` G ) ( A L b ) -> A = b ) )
57		idd 24	. . . . . . . 8 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) -> ( A = b -> A = b ) )
58	56, 57	jaod 378	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) -> ( ( D (⟂G ` G ) ( A L b ) \/ A = b ) -> A = b ) )
59	58	impr 617	. . . . . 6 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> A = b )
60	59	eqcomd 2462	. . . . 5 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> b = A )
61		simpr 459	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> b = A )
62	61	oveq2d 6286	. . . . . . . 8 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) b ) = ( A (midG ` G ) A ) )
63	5, 6, 7, 8, 10, 1, 1	midid 24351	. . . . . . . . 9 |- ( ph -> ( A (midG ` G ) A ) = A )
64	63	ad3antrrr 727	. . . . . . . 8 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) A ) = A )
65	62, 64	eqtrd 2495	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) b ) = A )
66		simpllr 758	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> A e. D )
67	65, 66	eqeltrd 2542	. . . . . 6 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) b ) e. D )
68	61	eqcomd 2462	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> A = b )
69	68	olcd 391	. . . . . 6 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( D (⟂G ` G ) ( A L b ) \/ A = b ) )
70	67, 69	jca 530	. . . . 5 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
71	60, 70	impbida 830	. . . 4 |- ( ( ( ph /\ A e. D ) /\ b e. P ) -> ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> b = A ) )
72	71	ralrimiva 2868	. . 3 |- ( ( ph /\ A e. D ) -> A. b e. P ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> b = A ) )
73		reu6i 3287	. . 3 |- ( ( A e. P /\ A. b e. P ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> b = A ) ) -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
74	2, 72, 73	syl2anc 659	. 2 |- ( ( ph /\ A e. D ) -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
75	8	adantr 463	. . . . . 6 |- ( ( ph /\ -. A e. D ) -> G e. TarskiG )
76	75	ad2antrr 723	. . . . 5 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> G e. TarskiG )
77	21	adantr 463	. . . . . . 7 |- ( ( ph /\ -. A e. D ) -> D e. ran L )
78	77	ad2antrr 723	. . . . . 6 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> D e. ran L )
79		simplr 753	. . . . . 6 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> x e. D )
80	5, 19, 7, 76, 78, 79	tglnpt 24140	. . . . 5 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> x e. P )
81		eqid 2454	. . . . 5 |- ( (pInvG ` G ) ` x ) = ( (pInvG ` G ) ` x )
82	1	adantr 463	. . . . . 6 |- ( ( ph /\ -. A e. D ) -> A e. P )
83	82	ad2antrr 723	. . . . 5 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> A e. P )
84	5, 6, 7, 19, 14, 76, 80, 81, 83	mircl 24246	. . . 4 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` x ) ` A ) e. P )
85		oveq2 6278	. . . . . . . . . 10 |- ( x = ( A (midG ` G ) b ) -> ( A L x ) = ( A L ( A (midG ` G ) b ) ) )
86	85	breq1d 4449	. . . . . . . . 9 |- ( x = ( A (midG ` G ) b ) -> ( ( A L x ) (⟂G ` G ) D <-> ( A L ( A (midG ` G ) b ) ) (⟂G ` G ) D ) )
87		simprl 754	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A (midG ` G ) b ) e. D )
88		simpr 459	. . . . . . . . . . . 12 |- ( ( ph /\ -. A e. D ) -> -. A e. D )
89	5, 6, 7, 19, 75, 77, 82, 88	foot 24300	. . . . . . . . . . 11 |- ( ( ph /\ -. A e. D ) -> E! x e. D ( A L x ) (⟂G ` G ) D )
90		reurmo 3072	. . . . . . . . . . 11 |- ( E! x e. D ( A L x ) (⟂G ` G ) D -> E* x e. D ( A L x ) (⟂G ` G ) D )
91	89, 90	syl 16	. . . . . . . . . 10 |- ( ( ph /\ -. A e. D ) -> E* x e. D ( A L x ) (⟂G ` G ) D )
92	91	ad4antr 729	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> E* x e. D ( A L x ) (⟂G ` G ) D )
93	79	ad2antrr 723	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> x e. D )
94		simpllr 758	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A L x ) (⟂G ` G ) D )
95	76	ad2antrr 723	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> G e. TarskiG )
96	83	ad2antrr 723	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> A e. P )
97		simplr 753	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> b e. P )
98	10	ad5antr 731	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> GDimTarskiG≥ 2 )
99	5, 6, 7, 95, 98, 96, 97	midcl 24347	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A (midG ` G ) b ) e. P )
100	5, 6, 7, 95, 98, 96, 97	midbtwn 24349	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A (midG ` G ) b ) e. ( A I b ) )
101	88	ad4antr 729	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> -. A e. D )
102		nelne2 2784	. . . . . . . . . . . . 13 |- ( ( ( A (midG ` G ) b ) e. D /\ -. A e. D ) -> ( A (midG ` G ) b ) =/= A )
103	87, 101, 102	syl2anc 659	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A (midG ` G ) b ) =/= A )
104	5, 6, 7, 95, 96, 99, 97, 100, 103	tgbtwnne 24085	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> A =/= b )
105	5, 7, 19, 95, 96, 97, 99, 104, 100	btwnlng1 24203	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A (midG ` G ) b ) e. ( A L b ) )
106	5, 7, 19, 95, 96, 97, 104, 99, 103, 105	tglineelsb2 24216	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A L b ) = ( A L ( A (midG ` G ) b ) ) )
107	78	ad2antrr 723	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> D e. ran L )
108	5, 7, 19, 95, 96, 97, 104	tgelrnln 24214	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A L b ) e. ran L )
109	104	neneqd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> -. A = b )
110		simprr 755	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( D (⟂G ` G ) ( A L b ) \/ A = b ) )
111	110	orcomd 386	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A = b \/ D (⟂G ` G ) ( A L b ) ) )
112	111	ord 375	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( -. A = b -> D (⟂G ` G ) ( A L b ) ) )
113	109, 112	mpd 15	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> D (⟂G ` G ) ( A L b ) )
114	5, 6, 7, 19, 95, 107, 108, 113	perpcom 24294	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A L b ) (⟂G ` G ) D )
115	106, 114	eqbrtrrd 4461	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A L ( A (midG ` G ) b ) ) (⟂G ` G ) D )
116	86, 87, 92, 93, 94, 115	rmoi2 3419	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> x = ( A (midG ` G ) b ) )
117	116	eqcomd 2462	. . . . . . 7 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( A (midG ` G ) b ) = x )
118	80	ad2antrr 723	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> x e. P )
119	5, 6, 7, 95, 98, 96, 97, 14, 118	ismidb 24348	. . . . . . 7 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( b = ( ( (pInvG ` G ) ` x ) ` A ) <-> ( A (midG ` G ) b ) = x ) )
120	117, 119	mpbird 232	. . . . . 6 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> b = ( ( (pInvG ` G ) ` x ) ` A ) )
121		simpr 459	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> b = ( ( (pInvG ` G ) ` x ) ` A ) )
122	76	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> G e. TarskiG )
123	10	ad5antr 731	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> GDimTarskiG≥ 2 )
124	83	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> A e. P )
125		simplr 753	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> b e. P )
126	80	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> x e. P )
127	5, 6, 7, 122, 123, 124, 125, 14, 126	ismidb 24348	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( b = ( ( (pInvG ` G ) ` x ) ` A ) <-> ( A (midG ` G ) b ) = x ) )
128	121, 127	mpbid 210	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( A (midG ` G ) b ) = x )
129	79	ad2antrr 723	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> x e. D )
130	128, 129	eqeltrd 2542	. . . . . . 7 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( A (midG ` G ) b ) e. D )
131	122	adantr 463	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> G e. TarskiG )
132		simp-4r 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> ( A L x ) (⟂G ` G ) D )
133	19, 131, 132	perpln1 24291	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> ( A L x ) e. ran L )
134	78	ad3antrrr 727	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> D e. ran L )
135	5, 6, 7, 19, 131, 133, 134, 132	perpcom 24294	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> D (⟂G ` G ) ( A L x ) )
136	124	adantr 463	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> A e. P )
137	126	adantr 463	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> x e. P )
138	5, 7, 19, 131, 136, 137, 133	tglnne 24212	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> A =/= x )
139		simpllr 758	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> b e. P )
140		simpr 459	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> A =/= b )
141	140	necomd 2725	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> b =/= A )
142	5, 6, 7, 19, 14, 131, 137, 81, 136	mirbtwn 24243	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> x e. ( ( ( (pInvG ` G ) ` x ) ` A ) I A ) )
143		simplr 753	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> b = ( ( (pInvG ` G ) ` x ) ` A ) )
144	143	oveq1d 6285	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> ( b I A ) = ( ( ( (pInvG ` G ) ` x ) ` A ) I A ) )
145	142, 144	eleqtrrd 2545	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> x e. ( b I A ) )
146	5, 7, 19, 131, 139, 136, 137, 141, 145	btwnlng1 24203	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> x e. ( b L A ) )
147	5, 7, 19, 131, 136, 137, 139, 138, 146, 141	lnrot1 24207	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> b e. ( A L x ) )
148	5, 7, 19, 131, 136, 137, 138, 139, 141, 147	tglineelsb2 24216	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> ( A L x ) = ( A L b ) )
149	135, 148	breqtrd 4463	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) /\ A =/= b ) -> D (⟂G ` G ) ( A L b ) )
150	149	ex 432	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( A =/= b -> D (⟂G ` G ) ( A L b ) ) )
151	150	necon1bd 2672	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( -. D (⟂G ` G ) ( A L b ) -> A = b ) )
152	151	orrd 376	. . . . . . 7 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( D (⟂G ` G ) ( A L b ) \/ A = b ) )
153	130, 152	jca 530	. . . . . 6 |- ( ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) /\ b = ( ( (pInvG ` G ) ` x ) ` A ) ) -> ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
154	120, 153	impbida 830	. . . . 5 |- ( ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ b e. P ) -> ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> b = ( ( (pInvG ` G ) ` x ) ` A ) ) )
155	154	ralrimiva 2868	. . . 4 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> A. b e. P ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> b = ( ( (pInvG ` G ) ` x ) ` A ) ) )
156		reu6i 3287	. . . 4 |- ( ( ( ( (pInvG ` G ) ` x ) ` A ) e. P /\ A. b e. P ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> b = ( ( (pInvG ` G ) ` x ) ` A ) ) ) -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
157	84, 155, 156	syl2anc 659	. . 3 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
158	5, 6, 7, 19, 75, 77, 82, 88	footex 24299	. . 3 |- ( ( ph /\ -. A e. D ) -> E. x e. D ( A L x ) (⟂G ` G ) D )
159	157, 158	r19.29a 2996	. 2 |- ( ( ph /\ -. A e. D ) -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
160	74, 159	pm2.61dan 789	1 |- ( ph -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E!wreu 2806   E*wrmo 2807   class class class wbr 4439   ran crn 4989   ` cfv 5570  (class class class)co 6270   2c2 10581   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  Dim_TarskiG≥cstrkgld 24030  Itvcitv 24033  LineGclng 24034  pInvGcmir 24237  ⟂Gcperpg 24276  midGcmid 24342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532  df-s2 12807  df-s3 12808  df-trkgc 24045  df-trkgb 24046  df-trkgcb 24047  df-trkgld 24049  df-trkg 24051  df-cgrg 24107  df-leg 24174  df-mir 24238  df-rag 24275  df-perpg 24277  df-mid 24344

This theorem is referenced by:  lmif  24355  islmib  24357