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	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	isinag.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	isinag.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	isinag.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	inagflat.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	inagflat.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	inagflat.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	inagflat.2	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
	inagflat.3	⊢ ( 𝜑 → 𝑋 ≠ 𝐵 )
	inagflat.4	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
Assertion	inagflat	⊢ ( 𝜑 → 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	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
5		isinag.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		isinag.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		isinag.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		inagflat.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
9		inagflat.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
10		inagflat.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
11		inagflat.2	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
12		inagflat.3	⊢ ( 𝜑 → 𝑋 ≠ 𝐵 )
13		inagflat.4	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
14		eqidd	⊢ ( 𝜑 → 𝐵 = 𝐵 )
15	14	orcd	⊢ ( 𝜑 → ( 𝐵 = 𝐵 ∨ 𝐵 ( 𝐾 ‘ 𝐵 ) 𝑋 ) )
16	1 2 3 4 5 6 7 8 6 10 11 12 13 15	isinagd	⊢ ( 𝜑 → 𝑋 ( inA ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ )