Theorem ragncol

Description: Right angle implies non-colinearity. A consequence of theorem 8.9 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 1-Dec-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 )
	ragncol.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
	ragncol.2	|- ( ph -> A =/= B )
	ragncol.3	|- ( ph -> C =/= B )
Assertion	ragncol	|- ( ph -> -. ( C e. ( A L B ) \/ A = 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		ragncol.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
11		ragncol.2	|- ( ph -> A =/= B )
12		ragncol.3	|- ( ph -> C =/= B )
13	11	neneqd	|- ( ph -> -. A = B )
14	12	neneqd	|- ( ph -> -. C = B )
15		ioran	|- ( -. ( A = B \/ C = B ) <-> ( -. A = B /\ -. C = B ) )
16	13 14 15	sylanbrc	|- ( ph -> -. ( A = B \/ C = B ) )
17	6	adantr	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> G e. TarskiG )
18	7	adantr	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> A e. P )
19	8	adantr	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> B e. P )
20	9	adantr	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> C e. P )
21	10	adantr	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> <" A B C "> e. ( raG ` G ) )
22		simpr	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> ( C e. ( A L B ) \/ A = B ) )
23	1 2 3 4 5 17 18 19 20 21 22	ragflat3	|- ( ( ph /\ ( C e. ( A L B ) \/ A = B ) ) -> ( A = B \/ C = B ) )
24	16 23	mtand	|- ( ph -> -. ( C e. ( A L B ) \/ A = B ) )