Theorem miduniq1

Description: Uniqueness of the middle point, expressed with point inversion. Theorem 7.18 of Schwabhauser p. 52. (Contributed by Thierry Arnoux, 30-Jul-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 )
	miduniq1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	miduniq1.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	miduniq1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	miduniq1.e	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐵 ) ‘ 𝑋 ) )
Assertion	miduniq1	⊢ ( 𝜑 → 𝐴 = 𝐵 )

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		miduniq1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		miduniq1.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
9		miduniq1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
10		miduniq1.e	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐵 ) ‘ 𝑋 ) )
11		eqid	⊢ ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐴 )
12	1 2 3 4 5 6 7 11 9	mircl	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) ∈ 𝑃 )
13		eqidd	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) )
14	10	eqcomd	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝐴 ) ‘ 𝑋 ) )
15	1 2 3 4 5 6 7 8 9 12 13 14	miduniq	⊢ ( 𝜑 → 𝐴 = 𝐵 )