Metamath Proof Explorer


Theorem lmif

Description: Line mirror as a function. (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 )
Assertion lmif ( 𝜑𝑀 : 𝑃𝑃 )

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 df-lmi lInvG = ( 𝑔 ∈ V ↦ ( 𝑑 ∈ ran ( LineG ‘ 𝑔 ) ↦ ( 𝑎 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑏 ∈ ( Base ‘ 𝑔 ) ( ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) )
10 fveq2 ( 𝑔 = 𝐺 → ( LineG ‘ 𝑔 ) = ( LineG ‘ 𝐺 ) )
11 10 7 eqtr4di ( 𝑔 = 𝐺 → ( LineG ‘ 𝑔 ) = 𝐿 )
12 11 rneqd ( 𝑔 = 𝐺 → ran ( LineG ‘ 𝑔 ) = ran 𝐿 )
13 fveq2 ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) )
14 13 1 eqtr4di ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝑃 )
15 fveq2 ( 𝑔 = 𝐺 → ( midG ‘ 𝑔 ) = ( midG ‘ 𝐺 ) )
16 15 oveqd ( 𝑔 = 𝐺 → ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) = ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) )
17 16 eleq1d ( 𝑔 = 𝐺 → ( ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) ∈ 𝑑 ↔ ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ) )
18 eqidd ( 𝑔 = 𝐺𝑑 = 𝑑 )
19 fveq2 ( 𝑔 = 𝐺 → ( ⟂G ‘ 𝑔 ) = ( ⟂G ‘ 𝐺 ) )
20 11 oveqd ( 𝑔 = 𝐺 → ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) = ( 𝑎 𝐿 𝑏 ) )
21 18 19 20 breq123d ( 𝑔 = 𝐺 → ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ↔ 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ) )
22 21 orbi1d ( 𝑔 = 𝐺 → ( ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ∨ 𝑎 = 𝑏 ) ↔ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) )
23 17 22 anbi12d ( 𝑔 = 𝐺 → ( ( ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ↔ ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) )
24 14 23 riotaeqbidv ( 𝑔 = 𝐺 → ( 𝑏 ∈ ( Base ‘ 𝑔 ) ( ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) = ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) )
25 14 24 mpteq12dv ( 𝑔 = 𝐺 → ( 𝑎 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑏 ∈ ( Base ‘ 𝑔 ) ( ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) = ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) )
26 12 25 mpteq12dv ( 𝑔 = 𝐺 → ( 𝑑 ∈ ran ( LineG ‘ 𝑔 ) ↦ ( 𝑎 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑏 ∈ ( Base ‘ 𝑔 ) ( ( 𝑎 ( midG ‘ 𝑔 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝑔 ) ( 𝑎 ( LineG ‘ 𝑔 ) 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) = ( 𝑑 ∈ ran 𝐿 ↦ ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) )
27 4 elexd ( 𝜑𝐺 ∈ V )
28 7 fvexi 𝐿 ∈ V
29 rnexg ( 𝐿 ∈ V → ran 𝐿 ∈ V )
30 mptexg ( ran 𝐿 ∈ V → ( 𝑑 ∈ ran 𝐿 ↦ ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) ∈ V )
31 28 29 30 mp2b ( 𝑑 ∈ ran 𝐿 ↦ ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) ∈ V
32 31 a1i ( 𝜑 → ( 𝑑 ∈ ran 𝐿 ↦ ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) ∈ V )
33 9 26 27 32 fvmptd3 ( 𝜑 → ( lInvG ‘ 𝐺 ) = ( 𝑑 ∈ ran 𝐿 ↦ ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ) )
34 eleq2 ( 𝑑 = 𝐷 → ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ↔ ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ) )
35 breq1 ( 𝑑 = 𝐷 → ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ↔ 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ) )
36 35 orbi1d ( 𝑑 = 𝐷 → ( ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ↔ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) )
37 34 36 anbi12d ( 𝑑 = 𝐷 → ( ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ↔ ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) )
38 37 riotabidv ( 𝑑 = 𝐷 → ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) = ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) )
39 38 mpteq2dv ( 𝑑 = 𝐷 → ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) = ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) )
40 39 adantl ( ( 𝜑𝑑 = 𝐷 ) → ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝑑 ∧ ( 𝑑 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) = ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) )
41 1 fvexi 𝑃 ∈ V
42 41 mptex ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ∈ V
43 42 a1i ( 𝜑 → ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) ∈ V )
44 33 40 8 43 fvmptd ( 𝜑 → ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 ) = ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) )
45 6 44 eqtrid ( 𝜑𝑀 = ( 𝑎𝑃 ↦ ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ) )
46 4 adantr ( ( 𝜑𝑎𝑃 ) → 𝐺 ∈ TarskiG )
47 5 adantr ( ( 𝜑𝑎𝑃 ) → 𝐺 DimTarskiG≥ 2 )
48 8 adantr ( ( 𝜑𝑎𝑃 ) → 𝐷 ∈ ran 𝐿 )
49 simpr ( ( 𝜑𝑎𝑃 ) → 𝑎𝑃 )
50 1 2 3 46 47 7 48 49 lmieu ( ( 𝜑𝑎𝑃 ) → ∃! 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) )
51 riotacl ( ∃! 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) → ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ∈ 𝑃 )
52 50 51 syl ( ( 𝜑𝑎𝑃 ) → ( 𝑏𝑃 ( ( 𝑎 ( midG ‘ 𝐺 ) 𝑏 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑎 𝐿 𝑏 ) ∨ 𝑎 = 𝑏 ) ) ) ∈ 𝑃 )
53 45 52 fmpt3d ( 𝜑𝑀 : 𝑃𝑃 )