Theorem inagflat

Description: Any point lies in a flat angle. (Contributed by Thierry Arnoux, 13-Feb-2023)

	Ref	Expression
Hypotheses	isinag.p	|- P = ( Base ` G )
	isinag.i	|- I = ( Itv ` G )
	isinag.k	|- K = ( hlG ` G )
	isinag.x	|- ( ph -> X e. P )
	isinag.a	|- ( ph -> A e. P )
	isinag.b	|- ( ph -> B e. P )
	isinag.c	|- ( ph -> C e. P )
	inagflat.g	|- ( ph -> G e. TarskiG )
	inagflat.x	|- ( ph -> X e. P )
	inagflat.1	|- ( ph -> A =/= B )
	inagflat.2	|- ( ph -> C =/= B )
	inagflat.3	|- ( ph -> X =/= B )
	inagflat.4	|- ( ph -> B e. ( A I C ) )
Assertion	inagflat	|- ( ph -> X ( inA ` G ) <" A B C "> )

Proof

Step	Hyp	Ref	Expression
1		isinag.p	|- P = ( Base ` G )
2		isinag.i	|- I = ( Itv ` G )
3		isinag.k	|- K = ( hlG ` G )
4		isinag.x	|- ( ph -> X e. P )
5		isinag.a	|- ( ph -> A e. P )
6		isinag.b	|- ( ph -> B e. P )
7		isinag.c	|- ( ph -> C e. P )
8		inagflat.g	|- ( ph -> G e. TarskiG )
9		inagflat.x	|- ( ph -> X e. P )
10		inagflat.1	|- ( ph -> A =/= B )
11		inagflat.2	|- ( ph -> C =/= B )
12		inagflat.3	|- ( ph -> X =/= B )
13		inagflat.4	|- ( ph -> B e. ( A I C ) )
14		eqidd	|- ( ph -> B = B )
15	14	orcd	|- ( ph -> ( B = B \/ B ( K ` B ) X ) )
16	1 2 3 4 5 6 7 8 6 10 11 12 13 15	isinagd	|- ( ph -> X ( inA ` G ) <" A B C "> )