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Theorem islmib 25724
Description: Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
Hypotheses
Ref Expression
ismid.p 𝑃 = (Base‘𝐺)
ismid.d = (dist‘𝐺)
ismid.i 𝐼 = (Itv‘𝐺)
ismid.g (𝜑𝐺 ∈ TarskiG)
ismid.1 (𝜑𝐺DimTarskiG≥2)
lmif.m 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmif.l 𝐿 = (LineG‘𝐺)
lmif.d (𝜑𝐷 ∈ ran 𝐿)
lmicl.1 (𝜑𝐴𝑃)
islmib.b (𝜑𝐵𝑃)
Assertion
Ref Expression
islmib (𝜑 → (𝐵 = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))

Proof of Theorem islmib
Dummy variables 𝑎 𝑏 𝑔 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmif.m . . . . 5 𝑀 = ((lInvG‘𝐺)‘𝐷)
2 df-lmi 25712 . . . . . . . 8 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))))))
32a1i 11 . . . . . . 7 (𝜑 → 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)))))))
4 fveq2 6229 . . . . . . . . . . 11 (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
5 lmif.l . . . . . . . . . . 11 𝐿 = (LineG‘𝐺)
64, 5syl6eqr 2703 . . . . . . . . . 10 (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
76rneqd 5385 . . . . . . . . 9 (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
8 fveq2 6229 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
9 ismid.p . . . . . . . . . . 11 𝑃 = (Base‘𝐺)
108, 9syl6eqr 2703 . . . . . . . . . 10 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
11 fveq2 6229 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (midG‘𝑔) = (midG‘𝐺))
1211oveqd 6707 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑎(midG‘𝑔)𝑏) = (𝑎(midG‘𝐺)𝑏))
1312eleq1d 2715 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
14 eqidd 2652 . . . . . . . . . . . . . 14 (𝑔 = 𝐺𝑑 = 𝑑)
15 fveq2 6229 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
166oveqd 6707 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (𝑎(LineG‘𝑔)𝑏) = (𝑎𝐿𝑏))
1714, 15, 16breq123d 4699 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ↔ 𝑑(⟂G‘𝐺)(𝑎𝐿𝑏)))
1817orbi1d 739 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
1913, 18anbi12d 747 . . . . . . . . . . 11 (𝑔 = 𝐺 → (((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)) ↔ ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
2010, 19riotaeqbidv 6654 . . . . . . . . . 10 (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)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
2110, 20mpteq12dv 4766 . . . . . . . . 9 (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)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
227, 21mpteq12dv 4766 . . . . . . . 8 (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)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
2322adantl 481 . . . . . . 7 ((𝜑𝑔 = 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)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
24 ismid.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
25 elex 3243 . . . . . . . 8 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
2624, 25syl 17 . . . . . . 7 (𝜑𝐺 ∈ V)
27 fvex 6239 . . . . . . . . . 10 (LineG‘𝐺) ∈ V
285, 27eqeltri 2726 . . . . . . . . 9 𝐿 ∈ V
29 rnexg 7140 . . . . . . . . 9 (𝐿 ∈ V → ran 𝐿 ∈ V)
30 mptexg 6525 . . . . . . . . 9 (ran 𝐿 ∈ V → (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V)
3128, 29, 30mp2b 10 . . . . . . . 8 (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V
3231a1i 11 . . . . . . 7 (𝜑 → (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V)
333, 23, 26, 32fvmptd 6327 . . . . . 6 (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
34 eleq2 2719 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝐷))
35 breq1 4688 . . . . . . . . . . 11 (𝑑 = 𝐷 → (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝑎𝐿𝑏)))
3635orbi1d 739 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
3734, 36anbi12d 747 . . . . . . . . 9 (𝑑 = 𝐷 → (((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)) ↔ ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
3837riotabidv 6653 . . . . . . . 8 (𝑑 = 𝐷 → (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))) = (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
3938mpteq2dv 4778 . . . . . . 7 (𝑑 = 𝐷 → (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) = (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
4039adantl 481 . . . . . 6 ((𝜑𝑑 = 𝐷) → (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) = (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
41 lmif.d . . . . . 6 (𝜑𝐷 ∈ ran 𝐿)
42 fvex 6239 . . . . . . . . 9 (Base‘𝐺) ∈ V
439, 42eqeltri 2726 . . . . . . . 8 𝑃 ∈ V
4443mptex 6527 . . . . . . 7 (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V
4544a1i 11 . . . . . 6 (𝜑 → (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V)
4633, 40, 41, 45fvmptd 6327 . . . . 5 (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
471, 46syl5eq 2697 . . . 4 (𝜑𝑀 = (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
48 oveq1 6697 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝑏))
4948eleq1d 2715 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷))
50 oveq1 6697 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎𝐿𝑏) = (𝐴𝐿𝑏))
5150breq2d 4697 . . . . . . . 8 (𝑎 = 𝐴 → (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
52 eqeq1 2655 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎 = 𝑏𝐴 = 𝑏))
5351, 52orbi12d 746 . . . . . . 7 (𝑎 = 𝐴 → ((𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
5449, 53anbi12d 747 . . . . . 6 (𝑎 = 𝐴 → (((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)) ↔ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
5554riotabidv 6653 . . . . 5 (𝑎 = 𝐴 → (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))) = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
5655adantl 481 . . . 4 ((𝜑𝑎 = 𝐴) → (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))) = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
57 lmicl.1 . . . 4 (𝜑𝐴𝑃)
58 ismid.d . . . . . 6 = (dist‘𝐺)
59 ismid.i . . . . . 6 𝐼 = (Itv‘𝐺)
60 ismid.1 . . . . . 6 (𝜑𝐺DimTarskiG≥2)
619, 58, 59, 24, 60, 5, 41, 57lmieu 25721 . . . . 5 (𝜑 → ∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
62 riotacl 6665 . . . . 5 (∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) → (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) ∈ 𝑃)
6361, 62syl 17 . . . 4 (𝜑 → (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) ∈ 𝑃)
6447, 56, 57, 63fvmptd 6327 . . 3 (𝜑 → (𝑀𝐴) = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
6564eqeq2d 2661 . 2 (𝜑 → (𝐵 = (𝑀𝐴) ↔ 𝐵 = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))))
66 islmib.b . . . 4 (𝜑𝐵𝑃)
67 oveq2 6698 . . . . . . 7 (𝑏 = 𝐵 → (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝐵))
6867eleq1d 2715 . . . . . 6 (𝑏 = 𝐵 → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝐵) ∈ 𝐷))
69 oveq2 6698 . . . . . . . 8 (𝑏 = 𝐵 → (𝐴𝐿𝑏) = (𝐴𝐿𝐵))
7069breq2d 4697 . . . . . . 7 (𝑏 = 𝐵 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
71 eqeq2 2662 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 = 𝑏𝐴 = 𝐵))
7270, 71orbi12d 746 . . . . . 6 (𝑏 = 𝐵 → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
7368, 72anbi12d 747 . . . . 5 (𝑏 = 𝐵 → (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
7473riota2 6673 . . . 4 ((𝐵𝑃 ∧ ∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ↔ (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) = 𝐵))
7566, 61, 74syl2anc 694 . . 3 (𝜑 → (((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ↔ (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) = 𝐵))
76 eqcom 2658 . . 3 (𝐵 = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) ↔ (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) = 𝐵)
7775, 76syl6bbr 278 . 2 (𝜑 → (((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ↔ 𝐵 = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))))
7865, 77bitr4d 271 1 (𝜑 → (𝐵 = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  ∃!wreu 2943  Vcvv 3231   class class class wbr 4685  cmpt 4762  ran crn 5144  cfv 5926  crio 6650  (class class class)co 6690  2c2 11108  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  DimTarskiGcstrkgld 25378  Itvcitv 25380  LineGclng 25381  ⟂Gcperpg 25635  midGcmid 25709  lInvGclmi 25710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkgld 25396  df-trkg 25397  df-cgrg 25451  df-leg 25523  df-mir 25593  df-rag 25634  df-perpg 25636  df-mid 25711  df-lmi 25712
This theorem is referenced by:  lmicom  25725  lmiinv  25729  lmimid  25731  lmiisolem  25733  hypcgrlem1  25736  hypcgrlem2  25737  lmiopp  25739  trgcopyeulem  25742
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