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Theorem tngngp 23263
Description: Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
tngngp.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngngp.x 𝑋 = (Base‘𝐺)
tngngp.m = (-g𝐺)
tngngp.z 0 = (0g𝐺)
Assertion
Ref Expression
tngngp (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
Distinct variable groups:   𝑥,𝑦,   𝑥,𝑁,𝑦   𝑥,𝑇,𝑦   𝑥,𝑋,𝑦   𝑥, 0 ,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem tngngp
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngngp.t . . . . 5 𝑇 = (𝐺 toNrmGrp 𝑁)
2 tngngp.x . . . . 5 𝑋 = (Base‘𝐺)
3 eqid 2824 . . . . 5 (dist‘𝑇) = (dist‘𝑇)
41, 2, 3tngngp2 23261 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
54simprbda 502 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6 simplr 768 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑇 ∈ NrmGrp)
7 simpr 488 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥𝑋)
82fvexi 6675 . . . . . . . . . . 11 𝑋 ∈ V
9 reex 10626 . . . . . . . . . . 11 ℝ ∈ V
10 fex2 7633 . . . . . . . . . . 11 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
118, 9, 10mp3an23 1450 . . . . . . . . . 10 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
1211ad2antrr 725 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 ∈ V)
131, 2tngbas 23250 . . . . . . . . 9 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
1412, 13syl 17 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑋 = (Base‘𝑇))
157, 14eleqtrd 2918 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥 ∈ (Base‘𝑇))
16 eqid 2824 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
17 eqid 2824 . . . . . . . 8 (norm‘𝑇) = (norm‘𝑇)
18 eqid 2824 . . . . . . . 8 (0g𝑇) = (0g𝑇)
1916, 17, 18nmeq0 23227 . . . . . . 7 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
206, 15, 19syl2anc 587 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
215adantr 484 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
22 simpll 766 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁:𝑋⟶ℝ)
231, 2, 9tngnm 23260 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
2421, 22, 23syl2anc 587 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 = (norm‘𝑇))
2524fveq1d 6663 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
2625eqeq1d 2826 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
27 tngngp.z . . . . . . . . 9 0 = (0g𝐺)
281, 27tng0 23252 . . . . . . . 8 (𝑁 ∈ V → 0 = (0g𝑇))
2912, 28syl 17 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 0 = (0g𝑇))
3029eqeq2d 2835 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 = 0𝑥 = (0g𝑇)))
3120, 26, 303bitr4d 314 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
32 simpllr 775 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑇 ∈ NrmGrp)
3315adantr 484 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥 ∈ (Base‘𝑇))
3414eleq2d 2901 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑦𝑋𝑦 ∈ (Base‘𝑇)))
3534biimpa 480 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦 ∈ (Base‘𝑇))
36 eqid 2824 . . . . . . . . 9 (-g𝑇) = (-g𝑇)
3716, 17, 36nmmtri 23231 . . . . . . . 8 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
3832, 33, 35, 37syl3anc 1368 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
39 tngngp.m . . . . . . . . . . 11 = (-g𝐺)
402, 14syl5eqr 2873 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (Base‘𝐺) = (Base‘𝑇))
41 eqid 2824 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
421, 41tngplusg 23251 . . . . . . . . . . . . 13 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
4312, 42syl 17 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (+g𝐺) = (+g𝑇))
4440, 43grpsubpropd 18204 . . . . . . . . . . 11 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (-g𝐺) = (-g𝑇))
4539, 44syl5eq 2871 . . . . . . . . . 10 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → = (-g𝑇))
4645oveqd 7166 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 𝑦) = (𝑥(-g𝑇)𝑦))
4724, 46fveq12d 6668 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4847adantr 484 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4924fveq1d 6663 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑦) = ((norm‘𝑇)‘𝑦))
5025, 49oveq12d 7167 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5150adantr 484 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5238, 48, 513brtr4d 5084 . . . . . 6 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5352ralrimiva 3177 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5431, 53jca 515 . . . 4 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
5554ralrimiva 3177 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
565, 55jca 515 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
57 simprl 770 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝐺 ∈ Grp)
58 simpl 486 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑁:𝑋⟶ℝ)
59 simpl 486 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6059ralimi 3155 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6160ad2antll 728 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
62 fveq2 6661 . . . . . . 7 (𝑥 = 𝑎 → (𝑁𝑥) = (𝑁𝑎))
6362eqeq1d 2826 . . . . . 6 (𝑥 = 𝑎 → ((𝑁𝑥) = 0 ↔ (𝑁𝑎) = 0))
64 eqeq1 2828 . . . . . 6 (𝑥 = 𝑎 → (𝑥 = 0𝑎 = 0 ))
6563, 64bibi12d 349 . . . . 5 (𝑥 = 𝑎 → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
6665rspccva 3608 . . . 4 ((∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
6761, 66sylan 583 . . 3 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
68 simpr 488 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
6968ralimi 3155 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
7069ad2antll 728 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
71 fvoveq1 7172 . . . . . . 7 (𝑥 = 𝑎 → (𝑁‘(𝑥 𝑦)) = (𝑁‘(𝑎 𝑦)))
7262oveq1d 7164 . . . . . . 7 (𝑥 = 𝑎 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑦)))
7371, 72breq12d 5065 . . . . . 6 (𝑥 = 𝑎 → ((𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
74 oveq2 7157 . . . . . . . 8 (𝑦 = 𝑏 → (𝑎 𝑦) = (𝑎 𝑏))
7574fveq2d 6665 . . . . . . 7 (𝑦 = 𝑏 → (𝑁‘(𝑎 𝑦)) = (𝑁‘(𝑎 𝑏)))
76 fveq2 6661 . . . . . . . 8 (𝑦 = 𝑏 → (𝑁𝑦) = (𝑁𝑏))
7776oveq2d 7165 . . . . . . 7 (𝑦 = 𝑏 → ((𝑁𝑎) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑏)))
7875, 77breq12d 5065 . . . . . 6 (𝑦 = 𝑏 → ((𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏))))
7973, 78rspc2va 3620 . . . . 5 (((𝑎𝑋𝑏𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8079ancoms 462 . . . 4 ((∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8170, 80sylan 583 . . 3 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
821, 2, 39, 27, 57, 58, 67, 81tngngpd 23262 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑇 ∈ NrmGrp)
8356, 82impbida 800 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480   class class class wbr 5052  wf 6339  cfv 6343  (class class class)co 7149  cr 10534  0cc0 10535   + caddc 10538  cle 10674  Basecbs 16483  +gcplusg 16565  distcds 16574  0gc0g 16713  Grpcgrp 18103  -gcsg 18105  Metcmet 20531  normcnm 23186  NrmGrpcngp 23187   toNrmGrp ctng 23188
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  ax-pre-sup 10613
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-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-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-plusg 16578  df-tset 16584  df-ds 16587  df-rest 16696  df-topn 16697  df-0g 16715  df-topgen 16717  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-sbg 18108  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-xms 22930  df-ms 22931  df-nm 23192  df-ngp 23193  df-tng 23194
This theorem is referenced by: (None)
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