Theorem ragcol

Description: The right angle property is independent of the choice of point on one side. Theorem 8.3 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 25-Aug-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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	israg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	israg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	israg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	ragcol.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	ragcol.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
	ragcol.2	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	ragcol.3	⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐵 𝐿 𝐷 ) ∨ 𝐵 = 𝐷 ) )
Assertion	ragcol	⊢ ( 𝜑 → ⟨“ 𝐷 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )

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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		israg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		israg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
9		israg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
10		ragcol.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
11		ragcol.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
12		ragcol.2	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
13		ragcol.3	⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐵 𝐿 𝐷 ) ∨ 𝐵 = 𝐷 ) )
14		eqid	⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 )
15		eqid	⊢ ( 𝑆 ‘ 𝐵 ) = ( 𝑆 ‘ 𝐵 )
16	1 2 3 4 5 6 8 15 9	mircl	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ∈ 𝑃 )
17	12	necomd	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )
18	1 2 3 4 5 6 8 15 9	mircgr	⊢ ( 𝜑 → ( 𝐵 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = ( 𝐵 − 𝐶 ) )
19	18	eqcomd	⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐵 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) )
20	1 2 3 4 5 6 7 8 9	israg	⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) )
21	11 20	mpbid	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) )
22	1 4 3 6 8 7 10 14 9 16 2 17 13 19 21	lncgr	⊢ ( 𝜑 → ( 𝐷 − 𝐶 ) = ( 𝐷 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) )
23	1 2 3 4 5 6 10 8 9	israg	⊢ ( 𝜑 → ( ⟨“ 𝐷 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐷 − 𝐶 ) = ( 𝐷 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) )
24	22 23	mpbird	⊢ ( 𝜑 → ⟨“ 𝐷 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )