Theorem lmieu 25676

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 481	. . 3 |- ( ( ph /\ A e. D ) -> A e. P )
3		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> -. A = b )
4		eqidd 2623	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> GDimTarskiG≥ 2 )
12	2	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> A e. P )
13		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> b e. P )
14		eqid 2622	. . . . . . . . . . . . . . . 16 |- (pInvG ` G ) = (pInvG ` G )
15	5, 6, 7, 9, 11, 12, 13	midcl 25669	. . . . . . . . . . . . . . . 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 25670	. . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> D e. ran L )
23	22	adantr 481	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> A =/= b )
27	26	adantr 481	. . . . . . . . . . . . . . . . 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 25525	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . 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 807	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> A e. D )
31	30	adantr 481	. . . . . . . . . . . . . . . . 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 25528	. . . . . . . . . . . . . . . . 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 3798	. . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . 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 25671	. . . . . . . . . . . . . . . . . . 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 25514	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 3798	. . . . . . . . . . . . . . . 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 25538	. . . . . . . . . . . . . . 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 6195	. . . . . . . . . . . . . 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 6193	. . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . 14 |- ( (pInvG ` G ) ` A ) = ( (pInvG ` G ) ` A )
43	5, 6, 7, 19, 14, 20, 24, 42	mircinv 25563	. . . . . . . . . . . . 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 2663	. . . . . . . . . . . 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 691	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> -. D =/= ( A L b ) )
46	8	ad5antr 770	. . . . . . . . . . . 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 770	. . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 15 |- ( -. D =/= ( A L b ) <-> D = ( A L b ) )
49	45, 48	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ ( A (midG ` G ) b ) e. D ) /\ -. A = b ) -> D = ( A L b ) )
50	49	adantr 481	. . . . . . . . . . . . 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 2702	. . . . . . . . . . . 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 477	. . . . . . . . . . . 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 25609	. . . . . . . . . . 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 691	. . . . . . . . . 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 450	. . . . . . . . 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 114	. . . . . . . 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 395	. . . . . . 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 649	. . . . . 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 2628	. . . . 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 477	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> b = A )
62	61	oveq2d 6666	. . . . . . . 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 25673	. . . . . . . . 9 |- ( ph -> ( A (midG ` G ) A ) = A )
64	63	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) A ) = A )
65	62, 64	eqtrd 2656	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) b ) = A )
66		simpllr 799	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> A e. D )
67	65, 66	eqeltrd 2701	. . . . . 6 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( A (midG ` G ) b ) e. D )
68	61	eqcomd 2628	. . . . . . 7 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> A = b )
69	68	olcd 408	. . . . . 6 |- ( ( ( ( ph /\ A e. D ) /\ b e. P ) /\ b = A ) -> ( D (⟂G ` G ) ( A L b ) \/ A = b ) )
70	67, 69	jca 554	. . . . 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 877	. . . 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 2966	. . 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 3397	. . 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 693	. 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 481	. . . . . 6 |- ( ( ph /\ -. A e. D ) -> G e. TarskiG )
76	75	ad2antrr 762	. . . . 5 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> G e. TarskiG )
77	21	adantr 481	. . . . . . 7 |- ( ( ph /\ -. A e. D ) -> D e. ran L )
78	77	ad2antrr 762	. . . . . 6 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> D e. ran L )
79		simplr 792	. . . . . 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 25444	. . . . 5 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> x e. P )
81		eqid 2622	. . . . 5 |- ( (pInvG ` G ) ` x ) = ( (pInvG ` G ) ` x )
82	1	adantr 481	. . . . . 6 |- ( ( ph /\ -. A e. D ) -> A e. P )
83	82	ad2antrr 762	. . . . 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 25556	. . . 4 |- ( ( ( ( ph /\ -. A e. D ) /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` x ) ` A ) e. P )
85		oveq2 6658	. . . . . . . . . 10 |- ( x = ( A (midG ` G ) b ) -> ( A L x ) = ( A L ( A (midG ` G ) b ) ) )
86	85	breq1d 4663	. . . . . . . . 9 |- ( x = ( A (midG ` G ) b ) -> ( ( A L x ) (⟂G ` G ) D <-> ( A L ( A (midG ` G ) b ) ) (⟂G ` G ) D ) )
87		simprl 794	. . . . . . . . 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 477	. . . . . . . . . . . 12 |- ( ( ph /\ -. A e. D ) -> -. A e. D )
89	5, 6, 7, 19, 75, 77, 82, 88	foot 25614	. . . . . . . . . . 11 |- ( ( ph /\ -. A e. D ) -> E! x e. D ( A L x ) (⟂G ` G ) D )
90		reurmo 3161	. . . . . . . . . . 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 17	. . . . . . . . . 10 |- ( ( ph /\ -. A e. D ) -> E* x e. D ( A L x ) (⟂G ` G ) D )
92	91	ad4antr 768	. . . . . . . . 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 762	. . . . . . . . 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 799	. . . . . . . . 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 762	. . . . . . . . . . 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 762	. . . . . . . . . . 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 792	. . . . . . . . . . 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 770	. . . . . . . . . . . . 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 25669	. . . . . . . . . . . 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 25671	. . . . . . . . . . . 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 768	. . . . . . . . . . . . 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 2891	. . . . . . . . . . . . 13 |- ( ( ( A (midG ` G ) b ) e. D /\ -. A e. D ) -> ( A (midG ` G ) b ) =/= A )
103	87, 101, 102	syl2anc 693	. . . . . . . . . . . 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 25385	. . . . . . . . . . 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 25514	. . . . . . . . . . 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 25527	. . . . . . . . . 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 762	. . . . . . . . . . 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 25525	. . . . . . . . . . 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 2799	. . . . . . . . . . . 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 796	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 392	. . . . . . . . . . . 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 25608	. . . . . . . . . 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 4677	. . . . . . . . 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 3532	. . . . . . . 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 2628	. . . . . . 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 762	. . . . . . . 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 25670	. . . . . . 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 247	. . . . . 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 477	. . . . . . . . 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 762	. . . . . . . . . 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 770	. . . . . . . . . 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 762	. . . . . . . . . 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 792	. . . . . . . . . 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 762	. . . . . . . . . 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 25670	. . . . . . . . 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 222	. . . . . . . 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 762	. . . . . . . 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 2701	. . . . . . 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 481	. . . . . . . . . . . 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 807	. . . . . . . . . . . . 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 25605	. . . . . . . . . . . 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 766	. . . . . . . . . . . 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 25608	. . . . . . . . . . 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 481	. . . . . . . . . . . 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 481	. . . . . . . . . . . 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 25523	. . . . . . . . . . . 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 799	. . . . . . . . . . . 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 477	. . . . . . . . . . . . 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 2849	. . . . . . . . . . . 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 25553	. . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . . 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 25514	. . . . . . . . . . . . 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 25518	. . . . . . . . . . . 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 25527	. . . . . . . . . . 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 4679	. . . . . . . . . 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 450	. . . . . . . . 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 2812	. . . . . . . 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 393	. . . . . . 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 554	. . . . . 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 877	. . . . 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 2966	. . . 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 3397	. . . 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 693	. . 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 25613	. . 3 |- ( ( ph /\ -. A e. D ) -> E. x e. D ( A L x ) (⟂G ` G ) D )
159	157, 158	r19.29a 3078	. 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 832	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 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E!wreu 2914   E*wrmo 2915   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   2c2 11070   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ⟂Gcperpg 25590  midGcmid 25664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591  df-mid 25666

This theorem is referenced by:  lmif  25677  islmib  25679