Theorem ragflat3

Description: Right angle and colinearity. Theorem 8.9 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019)

	Ref	Expression
Hypotheses	israg.p	|- P = ( Base ` G )
	israg.d	|- .- = ( dist ` G )
	israg.i	|- I = ( Itv ` G )
	israg.l	|- L = ( LineG ` G )
	israg.s	|- S = ( pInvG ` G )
	israg.g	|- ( ph -> G e. TarskiG )
	israg.a	|- ( ph -> A e. P )
	israg.b	|- ( ph -> B e. P )
	israg.c	|- ( ph -> C e. P )
	ragflat3.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
	ragflat3.2	|- ( ph -> ( C e. ( A L B ) \/ A = B ) )
Assertion	ragflat3	|- ( ph -> ( A = B \/ C = B ) )

Proof

Step	Hyp	Ref	Expression
1		israg.p	|- P = ( Base ` G )
2		israg.d	|- .- = ( dist ` G )
3		israg.i	|- I = ( Itv ` G )
4		israg.l	|- L = ( LineG ` G )
5		israg.s	|- S = ( pInvG ` G )
6		israg.g	|- ( ph -> G e. TarskiG )
7		israg.a	|- ( ph -> A e. P )
8		israg.b	|- ( ph -> B e. P )
9		israg.c	|- ( ph -> C e. P )
10		ragflat3.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
11		ragflat3.2	|- ( ph -> ( C e. ( A L B ) \/ A = B ) )
12	6	adantr	|- ( ( ph /\ -. A = B ) -> G e. TarskiG )
13	9	adantr	|- ( ( ph /\ -. A = B ) -> C e. P )
14	8	adantr	|- ( ( ph /\ -. A = B ) -> B e. P )
15	7	adantr	|- ( ( ph /\ -. A = B ) -> A e. P )
16	10	adantr	|- ( ( ph /\ -. A = B ) -> <" A B C "> e. ( raG ` G ) )
17		simpr	|- ( ( ph /\ -. A = B ) -> -. A = B )
18	17	neqned	|- ( ( ph /\ -. A = B ) -> A =/= B )
19	11	adantr	|- ( ( ph /\ -. A = B ) -> ( C e. ( A L B ) \/ A = B ) )
20	1 4 3 12 15 14 13 19	colrot1	|- ( ( ph /\ -. A = B ) -> ( A e. ( B L C ) \/ B = C ) )
21	1 2 3 4 5 12 15 14 13 13 16 18 20	ragcol	|- ( ( ph /\ -. A = B ) -> <" C B C "> e. ( raG ` G ) )
22	1 2 3 4 5 12 13 14 15 21	ragtriva	|- ( ( ph /\ -. A = B ) -> C = B )
23	22	ex	|- ( ph -> ( -. A = B -> C = B ) )
24	23	orrd	|- ( ph -> ( A = B \/ C = B ) )