Metamath Proof Explorer


Theorem lmireu

Description: Any point has a unique antecedent through line mirroring. Theorem 10.6 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 ( 𝜑𝐺 ∈ TarskiG )
ismid.1 ( 𝜑𝐺 DimTarskiG≥ 2 )
lmif.m 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
lmif.l 𝐿 = ( LineG ‘ 𝐺 )
lmif.d ( 𝜑𝐷 ∈ ran 𝐿 )
lmicl.1 ( 𝜑𝐴𝑃 )
Assertion lmireu ( 𝜑 → ∃! 𝑏𝑃 ( 𝑀𝑏 ) = 𝐴 )

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 ( 𝜑𝐺 ∈ TarskiG )
5 ismid.1 ( 𝜑𝐺 DimTarskiG≥ 2 )
6 lmif.m 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
7 lmif.l 𝐿 = ( LineG ‘ 𝐺 )
8 lmif.d ( 𝜑𝐷 ∈ ran 𝐿 )
9 lmicl.1 ( 𝜑𝐴𝑃 )
10 1 2 3 4 5 6 7 8 9 lmicl ( 𝜑 → ( 𝑀𝐴 ) ∈ 𝑃 )
11 1 2 3 4 5 6 7 8 9 lmilmi ( 𝜑 → ( 𝑀 ‘ ( 𝑀𝐴 ) ) = 𝐴 )
12 4 ad2antrr ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → 𝐺 ∈ TarskiG )
13 5 ad2antrr ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → 𝐺 DimTarskiG≥ 2 )
14 8 ad2antrr ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → 𝐷 ∈ ran 𝐿 )
15 simplr ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → 𝑏𝑃 )
16 1 2 3 12 13 6 7 14 15 lmilmi ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → ( 𝑀 ‘ ( 𝑀𝑏 ) ) = 𝑏 )
17 simpr ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → ( 𝑀𝑏 ) = 𝐴 )
18 17 fveq2d ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → ( 𝑀 ‘ ( 𝑀𝑏 ) ) = ( 𝑀𝐴 ) )
19 16 18 eqtr3d ( ( ( 𝜑𝑏𝑃 ) ∧ ( 𝑀𝑏 ) = 𝐴 ) → 𝑏 = ( 𝑀𝐴 ) )
20 19 ex ( ( 𝜑𝑏𝑃 ) → ( ( 𝑀𝑏 ) = 𝐴𝑏 = ( 𝑀𝐴 ) ) )
21 20 ralrimiva ( 𝜑 → ∀ 𝑏𝑃 ( ( 𝑀𝑏 ) = 𝐴𝑏 = ( 𝑀𝐴 ) ) )
22 fveqeq2 ( 𝑏 = ( 𝑀𝐴 ) → ( ( 𝑀𝑏 ) = 𝐴 ↔ ( 𝑀 ‘ ( 𝑀𝐴 ) ) = 𝐴 ) )
23 22 eqreu ( ( ( 𝑀𝐴 ) ∈ 𝑃 ∧ ( 𝑀 ‘ ( 𝑀𝐴 ) ) = 𝐴 ∧ ∀ 𝑏𝑃 ( ( 𝑀𝑏 ) = 𝐴𝑏 = ( 𝑀𝐴 ) ) ) → ∃! 𝑏𝑃 ( 𝑀𝑏 ) = 𝐴 )
24 10 11 21 23 syl3anc ( 𝜑 → ∃! 𝑏𝑃 ( 𝑀𝑏 ) = 𝐴 )