Theorem mirmot

Description: Point investion is a motion of the geometric space. Theorem 7.14 of Schwabhauser p. 51. (Contributed by Thierry Arnoux, 15-Dec-2019)

	Ref	Expression
Hypotheses	mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	mirval.d	⊢ − = ( dist ‘ 𝐺 )
	mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
	mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	mirmot.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
	mirmot.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
Assertion	mirmot	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		mirval.d	⊢ − = ( dist ‘ 𝐺 )
3		mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
6		mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		mirmot.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
8		mirmot.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
9	1 2 3 4 5 6 8 7	mirf1o	⊢ ( 𝜑 → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )
10	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐺 ∈ TarskiG )
11	8	adantr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐴 ∈ 𝑃 )
12		simprl	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝑎 ∈ 𝑃 )
13		simprr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝑏 ∈ 𝑃 )
14	1 2 3 4 5 10 11 7 12 13	miriso	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) )
15	14	ralrimivva	⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) )
16	1 2	ismot	⊢ ( 𝐺 ∈ TarskiG → ( 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( 𝑀 : 𝑃 –1-1-onto→ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) ) )
17	6 16	syl	⊢ ( 𝜑 → ( 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( 𝑀 : 𝑃 –1-1-onto→ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) ) )
18	9 15 17	mpbir2and	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) )