Theorem ragmir

Description: Right angle property is preserved by point inversion. Theorem 8.4 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 )
	ragmir.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
Assertion	ragmir	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 ( ( 𝑆 ‘ 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		ragmir.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
11		eqid	⊢ ( 𝑆 ‘ 𝐵 ) = ( 𝑆 ‘ 𝐵 )
12	1 2 3 4 5 6 8 11 9	mirmir	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = 𝐶 )
13	12	oveq2d	⊢ ( 𝜑 → ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) = ( 𝐴 − 𝐶 ) )
14	1 2 3 4 5 6 7 8 9	israg	⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) )
15	10 14	mpbid	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) )
16	13 15	eqtr2d	⊢ ( 𝜑 → ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) )
17	1 2 3 4 5 6 8 11 9	mircl	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ∈ 𝑃 )
18	1 2 3 4 5 6 7 8 17	israg	⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ”⟩ ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) ) )
19	16 18	mpbird	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ”⟩ ∈ ( ∟G ‘ 𝐺 ) )