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Theorem islmib 26584
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 26572 . . . . . . 7 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))))))
3 fveq2 6661 . . . . . . . . . 10 (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4 lmif.l . . . . . . . . . 10 𝐿 = (LineG‘𝐺)
53, 4syl6eqr 2877 . . . . . . . . 9 (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
65rneqd 5795 . . . . . . . 8 (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7 fveq2 6661 . . . . . . . . . 10 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8 ismid.p . . . . . . . . . 10 𝑃 = (Base‘𝐺)
97, 8syl6eqr 2877 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10 fveq2 6661 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (midG‘𝑔) = (midG‘𝐺))
1110oveqd 7166 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑎(midG‘𝑔)𝑏) = (𝑎(midG‘𝐺)𝑏))
1211eleq1d 2900 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
13 eqidd 2825 . . . . . . . . . . . . 13 (𝑔 = 𝐺𝑑 = 𝑑)
14 fveq2 6661 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
155oveqd 7166 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑎(LineG‘𝑔)𝑏) = (𝑎𝐿𝑏))
1613, 14, 15breq123d 5066 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ↔ 𝑑(⟂G‘𝐺)(𝑎𝐿𝑏)))
1716orbi1d 914 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
1812, 17anbi12d 633 . . . . . . . . . 10 (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))))
199, 18riotaeqbidv 7110 . . . . . . . . 9 (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))))
209, 19mpteq12dv 5137 . . . . . . . 8 (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)))))
216, 20mpteq12dv 5137 . . . . . . 7 (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))))))
22 ismid.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
2322elexd 3500 . . . . . . 7 (𝜑𝐺 ∈ V)
244fvexi 6675 . . . . . . . . 9 𝐿 ∈ V
25 rnexg 7609 . . . . . . . . 9 (𝐿 ∈ V → ran 𝐿 ∈ V)
26 mptexg 6975 . . . . . . . . 9 (ran 𝐿 ∈ V → (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V)
2724, 25, 26mp2b 10 . . . . . . . 8 (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V
2827a1i 11 . . . . . . 7 (𝜑 → (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V)
292, 21, 23, 28fvmptd3 6782 . . . . . 6 (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
30 eleq2 2904 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝐷))
31 breq1 5055 . . . . . . . . . . 11 (𝑑 = 𝐷 → (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝑎𝐿𝑏)))
3231orbi1d 914 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
3330, 32anbi12d 633 . . . . . . . . 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))))
3433riotabidv 7109 . . . . . . . 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))))
3534mpteq2dv 5148 . . . . . . 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)))))
3635adantl 485 . . . . . 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)))))
37 lmif.d . . . . . 6 (𝜑𝐷 ∈ ran 𝐿)
388fvexi 6675 . . . . . . . 8 𝑃 ∈ V
3938mptex 6977 . . . . . . 7 (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V
4039a1i 11 . . . . . 6 (𝜑 → (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V)
4129, 36, 37, 40fvmptd 6766 . . . . 5 (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
421, 41syl5eq 2871 . . . 4 (𝜑𝑀 = (𝑎𝑃 ↦ (𝑏𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
43 oveq1 7156 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝑏))
4443eleq1d 2900 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷))
45 oveq1 7156 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎𝐿𝑏) = (𝐴𝐿𝑏))
4645breq2d 5064 . . . . . . . 8 (𝑎 = 𝐴 → (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
47 eqeq1 2828 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎 = 𝑏𝐴 = 𝑏))
4846, 47orbi12d 916 . . . . . . 7 (𝑎 = 𝐴 → ((𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
4944, 48anbi12d 633 . . . . . 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))))
5049riotabidv 7109 . . . . 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))))
5150adantl 485 . . . 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))))
52 lmicl.1 . . . 4 (𝜑𝐴𝑃)
53 ismid.d . . . . . 6 = (dist‘𝐺)
54 ismid.i . . . . . 6 𝐼 = (Itv‘𝐺)
55 ismid.1 . . . . . 6 (𝜑𝐺DimTarskiG≥2)
568, 53, 54, 22, 55, 4, 37, 52lmieu 26581 . . . . 5 (𝜑 → ∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
57 riotacl 7124 . . . . 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)
5856, 57syl 17 . . . 4 (𝜑 → (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) ∈ 𝑃)
5942, 51, 52, 58fvmptd 6766 . . 3 (𝜑 → (𝑀𝐴) = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
6059eqeq2d 2835 . 2 (𝜑 → (𝐵 = (𝑀𝐴) ↔ 𝐵 = (𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))))
61 islmib.b . . . 4 (𝜑𝐵𝑃)
62 oveq2 7157 . . . . . . 7 (𝑏 = 𝐵 → (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝐵))
6362eleq1d 2900 . . . . . 6 (𝑏 = 𝐵 → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝐵) ∈ 𝐷))
64 oveq2 7157 . . . . . . . 8 (𝑏 = 𝐵 → (𝐴𝐿𝑏) = (𝐴𝐿𝐵))
6564breq2d 5064 . . . . . . 7 (𝑏 = 𝐵 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
66 eqeq2 2836 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 = 𝑏𝐴 = 𝐵))
6765, 66orbi12d 916 . . . . . 6 (𝑏 = 𝐵 → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
6863, 67anbi12d 633 . . . . 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))))
6968riota2 7132 . . . 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))
7061, 56, 69syl2anc 587 . . 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))
71 eqcom 2831 . . 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)
7270, 71syl6bbr 292 . 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)))))
7360, 72bitr4d 285 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 209  wa 399  wo 844   = wceq 1538  wcel 2115  ∃!wreu 3135  Vcvv 3480   class class class wbr 5052  cmpt 5132  ran crn 5543  cfv 6343  crio 7106  (class class class)co 7149  2c2 11689  Basecbs 16483  distcds 16574  TarskiGcstrkg 26227  DimTarskiGcstrkgld 26231  Itvcitv 26233  LineGclng 26234  ⟂Gcperpg 26492  midGcmid 26569  lInvGclmi 26570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-hash 13696  df-word 13867  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-trkgc 26245  df-trkgb 26246  df-trkgcb 26247  df-trkgld 26249  df-trkg 26250  df-cgrg 26308  df-leg 26380  df-mir 26450  df-rag 26491  df-perpg 26493  df-mid 26571  df-lmi 26572
This theorem is referenced by:  lmicom  26585  lmiinv  26589  lmimid  26591  lmiisolem  26593  hypcgrlem1  26596  hypcgrlem2  26597  lmiopp  26599  trgcopyeulem  26602
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