Theorem oppmir

Description: The mirror point with regard to a point X on a line A lies on the other side of A . (Contributed by Thierry Arnoux, 5-Jul-2026)

	Ref	Expression
Hypotheses	oppmir.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	oppmir.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	oppmir.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
	oppmir.m	⊢ 𝑀 = ( 𝑆 ‘ 𝑋 )
	oppmir.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
	oppmir.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	oppmir.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
	oppmir.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	oppmir.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑃 ∖ 𝐴 ) )
	oppmir.1	⊢ 𝐿 = ( LineG ‘ 𝐺 )
Assertion	oppmir	⊢ ( 𝜑 → 𝑌 𝑂 ( 𝑀 ‘ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		oppmir.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		oppmir.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		oppmir.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
4		oppmir.m	⊢ 𝑀 = ( 𝑆 ‘ 𝑋 )
5		oppmir.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
6		oppmir.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		oppmir.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
8		oppmir.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
9		oppmir.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑃 ∖ 𝐴 ) )
10		oppmir.1	⊢ 𝐿 = ( LineG ‘ 𝐺 )
11		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
12	9	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
13	1 10 2 6 7 8	tglnpt	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
14	1 11 2 10 3 6 13 4 12	mircl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ∈ 𝑃 )
15	9	eldifbd	⊢ ( 𝜑 → ¬ 𝑌 ∈ 𝐴 )
16	1 11 2 10 3 6 13 4 12	mirmir	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝑌 ) ) = 𝑌 )
17	16	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑌 ) ) = 𝑌 )
18	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → 𝐺 ∈ TarskiG )
19	7	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → 𝐴 ∈ ran 𝐿 )
20	8	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → 𝑋 ∈ 𝐴 )
21		simpr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 )
22	1 11 2 10 3 18 4 19 20 21	mirln	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑌 ) ) ∈ 𝐴 )
23	17 22	eqeltrrd	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 ) → 𝑌 ∈ 𝐴 )
24	15 23	mtand	⊢ ( 𝜑 → ¬ ( 𝑀 ‘ 𝑌 ) ∈ 𝐴 )
25	1 11 2 10 3 6 13 4 12	mirbtwn	⊢ ( 𝜑 → 𝑋 ∈ ( ( 𝑀 ‘ 𝑌 ) 𝐼 𝑌 ) )
26	1 11 2 6 14 13 12 25	tgbtwncom	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑌 𝐼 ( 𝑀 ‘ 𝑌 ) ) )
27	1 11 2 5 12 14 8 15 24 26	islnoppd	⊢ ( 𝜑 → 𝑌 𝑂 ( 𝑀 ‘ 𝑌 ) )