Theorem lmieq

Description: Equality deduction for line mirroring. Theorem 10.7 of Schwabhauser p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019)

	Ref	Expression
Hypotheses	ismid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	ismid.d	⊢ − = ( dist ‘ 𝐺 )
	ismid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	ismid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	ismid.1	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )
	lmif.m	⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
	lmif.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	lmif.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
	lmicl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	lmieq.c	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	lmieq.d	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ 𝐵 ) )
Assertion	lmieq	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ismid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ismid.d	⊢ − = ( dist ‘ 𝐺 )
3		ismid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		ismid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		ismid.1	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )
6		lmif.m	⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
7		lmif.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
8		lmif.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
9		lmicl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
10		lmieq.c	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
11		lmieq.d	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ 𝐵 ) )
12		fveqeq2	⊢ ( 𝑏 = 𝐴 → ( ( 𝑀 ‘ 𝑏 ) = ( 𝑀 ‘ 𝐵 ) ↔ ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ 𝐵 ) ) )
13		fveqeq2	⊢ ( 𝑏 = 𝐵 → ( ( 𝑀 ‘ 𝑏 ) = ( 𝑀 ‘ 𝐵 ) ↔ ( 𝑀 ‘ 𝐵 ) = ( 𝑀 ‘ 𝐵 ) ) )
14	1 2 3 4 5 6 7 8 10	lmicl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ 𝑃 )
15	1 2 3 4 5 6 7 8 14	lmireu	⊢ ( 𝜑 → ∃! 𝑏 ∈ 𝑃 ( 𝑀 ‘ 𝑏 ) = ( 𝑀 ‘ 𝐵 ) )
16		eqidd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( 𝑀 ‘ 𝐵 ) )
17	12 13 15 9 10 11 16	reu2eqd	⊢ ( 𝜑 → 𝐴 = 𝐵 )