Theorem midex 25629

Description: Existence of the midpoint, part Theorem 8.22 of [Schwabhauser] p. 64. Note that this proof requires a construction in 2 dimensions or more, i.e. it does not prove the existence of a midpoint in dimension 1, for a geometry restricted to a line. (Contributed by Thierry Arnoux, 25-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 )
mideu.s	|- S = (pInvG ` G )
mideu.1	|- ( ph -> A e. P )
mideu.2	|- ( ph -> B e. P )
mideu.3	|- ( ph -> GDimTarskiG≥ 2 )

Assertion

Ref	Expression
midex	|- ( ph -> E. x e. P B = ( ( S ` x ) ` A ) )

Distinct variable groups:   x, .-   x, A   x, B   x, G   x, I   x, L   x, P   x, S   ph, x

Proof of Theorem midex

Dummy variables p q s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mideu.1	. . . 4 |- ( ph -> A e. P )
2	1	adantr 481	. . 3 |- ( ( ph /\ A = B ) -> A e. P )
3		colperpex.p	. . . . 5 |- P = ( Base ` G )
4		colperpex.d	. . . . 5 |- .- = ( dist ` G )
5		colperpex.i	. . . . 5 |- I = (Itv ` G )
6		colperpex.l	. . . . 5 |- L = (LineG ` G )
7		mideu.s	. . . . 5 |- S = (pInvG ` G )
8		colperpex.g	. . . . . 6 |- ( ph -> G e. TarskiG )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> G e. TarskiG )
10		eqid 2622	. . . . 5 |- ( S ` A ) = ( S ` A )
11	3, 4, 5, 6, 7, 9, 2, 10	mircinv 25563	. . . 4 |- ( ( ph /\ A = B ) -> ( ( S ` A ) ` A ) = A )
12		simpr 477	. . . 4 |- ( ( ph /\ A = B ) -> A = B )
13	11, 12	eqtr2d 2657	. . 3 |- ( ( ph /\ A = B ) -> B = ( ( S ` A ) ` A ) )
14		fveq2 6191	. . . . . 6 |- ( x = A -> ( S ` x ) = ( S ` A ) )
15	14	fveq1d 6193	. . . . 5 |- ( x = A -> ( ( S ` x ) ` A ) = ( ( S ` A ) ` A ) )
16	15	eqeq2d 2632	. . . 4 |- ( x = A -> ( B = ( ( S ` x ) ` A ) <-> B = ( ( S ` A ) ` A ) ) )
17	16	rspcev 3309	. . 3 |- ( ( A e. P /\ B = ( ( S ` A ) ` A ) ) -> E. x e. P B = ( ( S ` x ) ` A ) )
18	2, 13, 17	syl2anc 693	. 2 |- ( ( ph /\ A = B ) -> E. x e. P B = ( ( S ` x ) ` A ) )
19	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> G e. TarskiG )
20	19	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> G e. TarskiG )
21	20	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> G e. TarskiG )
22	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> A e. P )
23	22	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> A e. P )
24	23	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> A e. P )
25		mideu.2	. . . . . . . . 9 |- ( ph -> B e. P )
26	25	adantr 481	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> B e. P )
27	26	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> B e. P )
28	27	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> B e. P )
29		simpr 477	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> A =/= B )
30	29	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> A =/= B )
31	30	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> A =/= B )
32		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> q e. P )
33	32	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> q e. P )
34		simp-4r 807	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> p e. P )
35		simpllr 799	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> t e. P )
36		simp-5r 809	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( B L q ) (⟂G ` G ) ( A L B ) )
37	6, 21, 36	perpln1 25605	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( B L q ) e. ran L )
38	3, 5, 6, 21, 24, 28, 31	tgelrnln 25525	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A L B ) e. ran L )
39	3, 4, 5, 6, 21, 37, 38, 36	perpcom 25608	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A L B ) (⟂G ` G ) ( B L q ) )
40	3, 5, 6, 21, 28, 33, 37	tglnne 25523	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> B =/= q )
41	3, 5, 6, 21, 28, 33, 40	tglinecom 25530	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( B L q ) = ( q L B ) )
42	39, 41	breqtrd 4679	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A L B ) (⟂G ` G ) ( q L B ) )
43		simplr 792	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) )
44	43	simpld 475	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A L p ) (⟂G ` G ) ( A L B ) )
45	6, 21, 44	perpln1 25605	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A L p ) e. ran L )
46	3, 4, 5, 6, 21, 45, 38, 44	perpcom 25608	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A L B ) (⟂G ` G ) ( A L p ) )
47	31	neneqd 2799	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> -. A = B )
48	43	simprd 479	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) )
49	48	simpld 475	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( t e. ( A L B ) \/ A = B ) )
50	49	orcomd 403	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A = B \/ t e. ( A L B ) ) )
51	50	ord 392	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( -. A = B -> t e. ( A L B ) ) )
52	47, 51	mpd 15	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> t e. ( A L B ) )
53	48	simprd 479	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> t e. ( q I p ) )
54		simpr 477	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> ( A .- p ) (≤G ` G ) ( B .- q ) )
55	3, 4, 5, 6, 21, 7, 24, 28, 31, 33, 34, 35, 42, 46, 52, 53, 54	mideulem 25628	. . . . 5 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( A .- p ) (≤G ` G ) ( B .- q ) ) -> E. x e. P B = ( ( S ` x ) ` A ) )
56	20	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> G e. TarskiG )
57	56	adantr 481	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> G e. TarskiG )
58		simprl 794	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> x e. P )
59		eqid 2622	. . . . . . . 8 |- ( S ` x ) = ( S ` x )
60	27	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> B e. P )
61	60	adantr 481	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> B e. P )
62		simprr 796	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> A = ( ( S ` x ) ` B ) )
63	62	eqcomd 2628	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> ( ( S ` x ) ` B ) = A )
64	3, 4, 5, 6, 7, 57, 58, 59, 61, 63	mircom 25558	. . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> ( ( S ` x ) ` A ) = B )
65	64	eqcomd 2628	. . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) /\ ( x e. P /\ A = ( ( S ` x ) ` B ) ) ) -> B = ( ( S ` x ) ` A ) )
66	23	ad4antr 768	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> A e. P )
67	30	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> A =/= B )
68	67	necomd 2849	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> B =/= A )
69		simp-4r 807	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> p e. P )
70	32	ad4antr 768	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> q e. P )
71		simpllr 799	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> t e. P )
72		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) )
73	72	simpld 475	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( A L p ) (⟂G ` G ) ( A L B ) )
74	6, 56, 73	perpln1 25605	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( A L p ) e. ran L )
75	3, 5, 6, 56, 66, 69, 74	tglnne 25523	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> A =/= p )
76	3, 5, 6, 56, 66, 69, 75	tglinecom 25530	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( A L p ) = ( p L A ) )
77	76	eqcomd 2628	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( p L A ) = ( A L p ) )
78	77, 74	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( p L A ) e. ran L )
79	3, 5, 6, 56, 60, 66, 68	tgelrnln 25525	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B L A ) e. ran L )
80	3, 5, 6, 56, 66, 60, 67	tglinecom 25530	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( A L B ) = ( B L A ) )
81	73, 76, 80	3brtr3d 4684	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( p L A ) (⟂G ` G ) ( B L A ) )
82	3, 4, 5, 6, 56, 78, 79, 81	perpcom 25608	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B L A ) (⟂G ` G ) ( p L A ) )
83		simp-5r 809	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B L q ) (⟂G ` G ) ( A L B ) )
84	6, 56, 83	perpln1 25605	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B L q ) e. ran L )
85	83, 80	breqtrd 4679	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B L q ) (⟂G ` G ) ( B L A ) )
86	3, 4, 5, 6, 56, 84, 79, 85	perpcom 25608	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B L A ) (⟂G ` G ) ( B L q ) )
87	67	neneqd 2799	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> -. A = B )
88	72	simprd 479	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) )
89	88	simpld 475	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( t e. ( A L B ) \/ A = B ) )
90	89	orcomd 403	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( A = B \/ t e. ( A L B ) ) )
91	90	ord 392	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( -. A = B -> t e. ( A L B ) ) )
92	87, 91	mpd 15	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> t e. ( A L B ) )
93	92, 80	eleqtrd 2703	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> t e. ( B L A ) )
94	88	simprd 479	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> t e. ( q I p ) )
95	3, 4, 5, 56, 70, 71, 69, 94	tgbtwncom 25383	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> t e. ( p I q ) )
96		simpr 477	. . . . . . 7 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> ( B .- q ) (≤G ` G ) ( A .- p ) )
97	3, 4, 5, 6, 56, 7, 60, 66, 68, 69, 70, 71, 82, 86, 93, 95, 96	mideulem 25628	. . . . . 6 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> E. x e. P A = ( ( S ` x ) ` B ) )
98	65, 97	reximddv 3018	. . . . 5 |- ( ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) /\ ( B .- q ) (≤G ` G ) ( A .- p ) ) -> E. x e. P B = ( ( S ` x ) ` A ) )
99		eqid 2622	. . . . . 6 |- (≤G ` G ) = (≤G ` G )
100	20	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> G e. TarskiG )
101	23	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> A e. P )
102		simpllr 799	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> p e. P )
103	27	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> B e. P )
104	32	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> q e. P )
105	3, 4, 5, 99, 100, 101, 102, 103, 104	legtrid 25486	. . . . 5 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> ( ( A .- p ) (≤G ` G ) ( B .- q ) \/ ( B .- q ) (≤G ` G ) ( A .- p ) ) )
106	55, 98, 105	mpjaodan 827	. . . 4 |- ( ( ( ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) /\ p e. P ) /\ t e. P ) /\ ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) ) -> E. x e. P B = ( ( S ` x ) ` A ) )
107		mideu.3	. . . . . . . 8 |- ( ph -> GDimTarskiG≥ 2 )
108	107	adantr 481	. . . . . . 7 |- ( ( ph /\ A =/= B ) -> GDimTarskiG≥ 2 )
109	108	ad2antrr 762	. . . . . 6 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> GDimTarskiG≥ 2 )
110	3, 4, 5, 6, 20, 23, 27, 32, 30, 109	colperpex 25625	. . . . 5 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂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. ( q I p ) ) ) )
111		r19.42v 3092	. . . . . 6 |- ( E. t e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) <-> ( ( A L p ) (⟂G ` G ) ( A L B ) /\ E. t e. P ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) )
112	111	rexbii 3041	. . . . 5 |- ( E. p e. P E. t e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) <-> 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. ( q I p ) ) ) )
113	110, 112	sylibr 224	. . . 4 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> E. p e. P E. t e. P ( ( A L p ) (⟂G ` G ) ( A L B ) /\ ( ( t e. ( A L B ) \/ A = B ) /\ t e. ( q I p ) ) ) )
114	106, 113	r19.29vva 3081	. . 3 |- ( ( ( ( ph /\ A =/= B ) /\ q e. P ) /\ ( B L q ) (⟂G ` G ) ( A L B ) ) -> E. x e. P B = ( ( S ` x ) ` A ) )
115	29	necomd 2849	. . . . 5 |- ( ( ph /\ A =/= B ) -> B =/= A )
116	3, 4, 5, 6, 19, 26, 22, 22, 115, 108	colperpex 25625	. . . 4 |- ( ( ph /\ A =/= B ) -> E. q e. P ( ( B L q ) (⟂G ` G ) ( B L A ) /\ E. s e. P ( ( s e. ( B L A ) \/ B = A ) /\ s e. ( A I q ) ) ) )
117		simprl 794	. . . . . . 7 |- ( ( ( ph /\ A =/= B ) /\ ( ( B L q ) (⟂G ` G ) ( B L A ) /\ E. s e. P ( ( s e. ( B L A ) \/ B = A ) /\ s e. ( A I q ) ) ) ) -> ( B L q ) (⟂G ` G ) ( B L A ) )
118	3, 5, 6, 19, 22, 26, 29	tglinecom 25530	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> ( A L B ) = ( B L A ) )
119	118	adantr 481	. . . . . . 7 |- ( ( ( ph /\ A =/= B ) /\ ( ( B L q ) (⟂G ` G ) ( B L A ) /\ E. s e. P ( ( s e. ( B L A ) \/ B = A ) /\ s e. ( A I q ) ) ) ) -> ( A L B ) = ( B L A ) )
120	117, 119	breqtrrd 4681	. . . . . 6 |- ( ( ( ph /\ A =/= B ) /\ ( ( B L q ) (⟂G ` G ) ( B L A ) /\ E. s e. P ( ( s e. ( B L A ) \/ B = A ) /\ s e. ( A I q ) ) ) ) -> ( B L q ) (⟂G ` G ) ( A L B ) )
121	120	ex 450	. . . . 5 |- ( ( ph /\ A =/= B ) -> ( ( ( B L q ) (⟂G ` G ) ( B L A ) /\ E. s e. P ( ( s e. ( B L A ) \/ B = A ) /\ s e. ( A I q ) ) ) -> ( B L q ) (⟂G ` G ) ( A L B ) ) )
122	121	reximdv 3016	. . . 4 |- ( ( ph /\ A =/= B ) -> ( E. q e. P ( ( B L q ) (⟂G ` G ) ( B L A ) /\ E. s e. P ( ( s e. ( B L A ) \/ B = A ) /\ s e. ( A I q ) ) ) -> E. q e. P ( B L q ) (⟂G ` G ) ( A L B ) ) )
123	116, 122	mpd 15	. . 3 |- ( ( ph /\ A =/= B ) -> E. q e. P ( B L q ) (⟂G ` G ) ( A L B ) )
124	114, 123	r19.29a 3078	. 2 |- ( ( ph /\ A =/= B ) -> E. x e. P B = ( ( S ` x ) ` A ) )
125	18, 124	pm2.61dane 2881	1 |- ( ph -> E. x e. P B = ( ( S ` x ) ` A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   2c2 11070   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  ≤Gcleg 25477  pInvGcmir 25547  ⟂Gcperpg 25590

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

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

This theorem is referenced by:  mideu  25630  opphllem5  25643  opphl  25646