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Theorem tngngp 22866
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 2777 . . . . 5 (dist‘𝑇) = (dist‘𝑇)
41, 2, 3tngngp2 22864 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
54simprbda 494 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6 simplr 759 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑇 ∈ NrmGrp)
7 simpr 479 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥𝑋)
82fvexi 6460 . . . . . . . . . . 11 𝑋 ∈ V
9 reex 10363 . . . . . . . . . . 11 ℝ ∈ V
10 fex2 7400 . . . . . . . . . . 11 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
118, 9, 10mp3an23 1526 . . . . . . . . . 10 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
1211ad2antrr 716 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 ∈ V)
131, 2tngbas 22853 . . . . . . . . 9 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
1412, 13syl 17 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑋 = (Base‘𝑇))
157, 14eleqtrd 2860 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥 ∈ (Base‘𝑇))
16 eqid 2777 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
17 eqid 2777 . . . . . . . 8 (norm‘𝑇) = (norm‘𝑇)
18 eqid 2777 . . . . . . . 8 (0g𝑇) = (0g𝑇)
1916, 17, 18nmeq0 22830 . . . . . . 7 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
206, 15, 19syl2anc 579 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
215adantr 474 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
22 simpll 757 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁:𝑋⟶ℝ)
231, 2, 9tngnm 22863 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
2421, 22, 23syl2anc 579 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 = (norm‘𝑇))
2524fveq1d 6448 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
2625eqeq1d 2779 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
27 tngngp.z . . . . . . . . 9 0 = (0g𝐺)
281, 27tng0 22855 . . . . . . . 8 (𝑁 ∈ V → 0 = (0g𝑇))
2912, 28syl 17 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 0 = (0g𝑇))
3029eqeq2d 2787 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 = 0𝑥 = (0g𝑇)))
3120, 26, 303bitr4d 303 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
32 simpllr 766 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑇 ∈ NrmGrp)
3315adantr 474 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥 ∈ (Base‘𝑇))
3414eleq2d 2844 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑦𝑋𝑦 ∈ (Base‘𝑇)))
3534biimpa 470 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦 ∈ (Base‘𝑇))
36 eqid 2777 . . . . . . . . 9 (-g𝑇) = (-g𝑇)
3716, 17, 36nmmtri 22834 . . . . . . . 8 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
3832, 33, 35, 37syl3anc 1439 . . . . . . 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 2827 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (Base‘𝐺) = (Base‘𝑇))
41 eqid 2777 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
421, 41tngplusg 22854 . . . . . . . . . . . . 13 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
4312, 42syl 17 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (+g𝐺) = (+g𝑇))
4440, 43grpsubpropd 17907 . . . . . . . . . . 11 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (-g𝐺) = (-g𝑇))
4539, 44syl5eq 2825 . . . . . . . . . 10 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → = (-g𝑇))
4645oveqd 6939 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 𝑦) = (𝑥(-g𝑇)𝑦))
4724, 46fveq12d 6453 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4847adantr 474 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4924fveq1d 6448 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑦) = ((norm‘𝑇)‘𝑦))
5025, 49oveq12d 6940 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5150adantr 474 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5238, 48, 513brtr4d 4918 . . . . . 6 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5352ralrimiva 3147 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5431, 53jca 507 . . . 4 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
5554ralrimiva 3147 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
565, 55jca 507 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
57 simprl 761 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝐺 ∈ Grp)
58 simpl 476 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑁:𝑋⟶ℝ)
59 simpl 476 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6059ralimi 3133 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6160ad2antll 719 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
62 fveq2 6446 . . . . . . 7 (𝑥 = 𝑎 → (𝑁𝑥) = (𝑁𝑎))
6362eqeq1d 2779 . . . . . 6 (𝑥 = 𝑎 → ((𝑁𝑥) = 0 ↔ (𝑁𝑎) = 0))
64 eqeq1 2781 . . . . . 6 (𝑥 = 𝑎 → (𝑥 = 0𝑎 = 0 ))
6563, 64bibi12d 337 . . . . 5 (𝑥 = 𝑎 → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
6665rspccva 3509 . . . 4 ((∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
6761, 66sylan 575 . . 3 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
68 simpr 479 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
6968ralimi 3133 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
7069ad2antll 719 . . . 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 6945 . . . . . . 7 (𝑥 = 𝑎 → (𝑁‘(𝑥 𝑦)) = (𝑁‘(𝑎 𝑦)))
7262oveq1d 6937 . . . . . . 7 (𝑥 = 𝑎 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑦)))
7371, 72breq12d 4899 . . . . . 6 (𝑥 = 𝑎 → ((𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
74 oveq2 6930 . . . . . . . 8 (𝑦 = 𝑏 → (𝑎 𝑦) = (𝑎 𝑏))
7574fveq2d 6450 . . . . . . 7 (𝑦 = 𝑏 → (𝑁‘(𝑎 𝑦)) = (𝑁‘(𝑎 𝑏)))
76 fveq2 6446 . . . . . . . 8 (𝑦 = 𝑏 → (𝑁𝑦) = (𝑁𝑏))
7776oveq2d 6938 . . . . . . 7 (𝑦 = 𝑏 → ((𝑁𝑎) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑏)))
7875, 77breq12d 4899 . . . . . 6 (𝑦 = 𝑏 → ((𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏))))
7973, 78rspc2va 3524 . . . . 5 (((𝑎𝑋𝑏𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8079ancoms 452 . . . 4 ((∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8170, 80sylan 575 . . 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 22865 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑇 ∈ NrmGrp)
8356, 82impbida 791 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 198  wa 386   = wceq 1601  wcel 2106  wral 3089  Vcvv 3397   class class class wbr 4886  wf 6131  cfv 6135  (class class class)co 6922  cr 10271  0cc0 10272   + caddc 10275  cle 10412  Basecbs 16255  +gcplusg 16338  distcds 16347  0gc0g 16486  Grpcgrp 17809  -gcsg 17811  Metcmet 20128  normcnm 22789  NrmGrpcngp 22790   toNrmGrp ctng 22791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-sup 8636  df-inf 8637  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-plusg 16351  df-tset 16357  df-ds 16360  df-rest 16469  df-topn 16470  df-0g 16488  df-topgen 16490  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-sbg 17814  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-xms 22533  df-ms 22534  df-nm 22795  df-ngp 22796  df-tng 22797
This theorem is referenced by: (None)
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