Theorem motrag

Description: Right angles are preserved by motions. (Contributed by Thierry Arnoux, 16-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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	israg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	israg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	israg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	motrag.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
	motrag.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
Assertion	motrag	⊢ ( 𝜑 → ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ( 𝐹 ‘ 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		motrag.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
11		motrag.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
12		eqid	⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 )
13	1 2 6 10 7	motcl	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝑃 )
14	1 2 6 10 8	motcl	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ 𝑃 )
15	1 2 6 10 9	motcl	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ 𝑃 )
16		eqidd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
17		eqidd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )
18		eqidd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐶 ) )
19	1 2 12 6 7 8 9 16 17 18 10	motcgr3	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ( 𝐹 ‘ 𝐶 ) ”⟩ )
20	1 2 3 4 5 6 7 8 9 12 13 14 15 11 19	ragcgr	⊢ ( 𝜑 → ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ( 𝐹 ‘ 𝐶 ) ”⟩ ∈ ( ∟G ‘ 𝐺 ) )