Theorem mirne

Description: Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020)

	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 )
	mirinv.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	mirne.1	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )
Assertion	mirne	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ≠ 𝐴 )

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		mirinv.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
10		mirne.1	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )
11		simpr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → ( 𝑀 ‘ 𝐵 ) = 𝐴 )
12	11	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = ( 𝑀 ‘ 𝐴 ) )
13	1 2 3 4 5 6 7 8 9	mirmir	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = 𝐵 )
14	13	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = 𝐵 )
15		eqid	⊢ 𝐴 = 𝐴
16	1 2 3 4 5 6 7 8 7	mirinv	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) = 𝐴 ↔ 𝐴 = 𝐴 ) )
17	15 16	mpbiri	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = 𝐴 )
18	17	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → ( 𝑀 ‘ 𝐴 ) = 𝐴 )
19	12 14 18	3eqtr3d	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → 𝐵 = 𝐴 )
20	10	adantr	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → 𝐵 ≠ 𝐴 )
21	20	neneqd	⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐴 ) → ¬ 𝐵 = 𝐴 )
22	19 21	pm2.65da	⊢ ( 𝜑 → ¬ ( 𝑀 ‘ 𝐵 ) = 𝐴 )
23	22	neqned	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ≠ 𝐴 )