Theorem mirfv

Description: Value of the point inversion function M . Definition 7.5 of Schwabhauser p. 49. (Contributed by Thierry Arnoux, 30-May-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 )
	mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
	mirfv.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
Assertion	mirfv	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 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		mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
9		mirfv.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
10	1 2 3 4 5 6 7	mirval	⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) )
11	8 10	eqtrid	⊢ ( 𝜑 → 𝑀 = ( 𝑦 ∈ 𝑃 ↦ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) ) )
12		simplr	⊢ ( ( ( 𝜑 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 ∈ 𝑃 ) → 𝑦 = 𝐵 )
13	12	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 ∈ 𝑃 ) → ( 𝐴 − 𝑦 ) = ( 𝐴 − 𝐵 ) )
14	13	eqeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 ∈ 𝑃 ) → ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ↔ ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ) )
15	12	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 ∈ 𝑃 ) → ( 𝑧 𝐼 𝑦 ) = ( 𝑧 𝐼 𝐵 ) )
16	15	eleq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 ∈ 𝑃 ) → ( 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ↔ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) )
17	14 16	anbi12d	⊢ ( ( ( 𝜑 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 ∈ 𝑃 ) → ( ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ↔ ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) )
18	17	riotabidva	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝑦 ) ) ) = ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) )
19		riotaex	⊢ ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) ∈ V
20	19	a1i	⊢ ( 𝜑 → ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) ∈ V )
21	11 18 9 20	fvmptd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( ℩ 𝑧 ∈ 𝑃 ( ( 𝐴 − 𝑧 ) = ( 𝐴 − 𝐵 ) ∧ 𝐴 ∈ ( 𝑧 𝐼 𝐵 ) ) ) )