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Theorem tngngp 22835
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 2825 . . . . 5 (dist‘𝑇) = (dist‘𝑇)
41, 2, 3tngngp2 22833 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
54simprbda 494 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6 simplr 785 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑇 ∈ NrmGrp)
7 simpr 479 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥𝑋)
82fvexi 6451 . . . . . . . . . . 11 𝑋 ∈ V
9 reex 10350 . . . . . . . . . . 11 ℝ ∈ V
10 fex2 7388 . . . . . . . . . . 11 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
118, 9, 10mp3an23 1581 . . . . . . . . . 10 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
1211ad2antrr 717 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 ∈ V)
131, 2tngbas 22822 . . . . . . . . 9 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
1412, 13syl 17 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑋 = (Base‘𝑇))
157, 14eleqtrd 2908 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥 ∈ (Base‘𝑇))
16 eqid 2825 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
17 eqid 2825 . . . . . . . 8 (norm‘𝑇) = (norm‘𝑇)
18 eqid 2825 . . . . . . . 8 (0g𝑇) = (0g𝑇)
1916, 17, 18nmeq0 22799 . . . . . . 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 783 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁:𝑋⟶ℝ)
231, 2, 9tngnm 22832 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
2421, 22, 23syl2anc 579 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 = (norm‘𝑇))
2524fveq1d 6439 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
2625eqeq1d 2827 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
27 tngngp.z . . . . . . . . 9 0 = (0g𝐺)
281, 27tng0 22824 . . . . . . . 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 303 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
32 simpllr 793 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑇 ∈ NrmGrp)
3315adantr 474 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥 ∈ (Base‘𝑇))
3414eleq2d 2892 . . . . . . . . 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 2825 . . . . . . . . 9 (-g𝑇) = (-g𝑇)
3716, 17, 36nmmtri 22803 . . . . . . . 8 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
3832, 33, 35, 37syl3anc 1494 . . . . . . 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 2875 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (Base‘𝐺) = (Base‘𝑇))
41 eqid 2825 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
421, 41tngplusg 22823 . . . . . . . . . . . . 13 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
4312, 42syl 17 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (+g𝐺) = (+g𝑇))
4440, 43grpsubpropd 17881 . . . . . . . . . . 11 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (-g𝐺) = (-g𝑇))
4539, 44syl5eq 2873 . . . . . . . . . 10 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → = (-g𝑇))
4645oveqd 6927 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 𝑦) = (𝑥(-g𝑇)𝑦))
4724, 46fveq12d 6444 . . . . . . . 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 6439 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑦) = ((norm‘𝑇)‘𝑦))
5025, 49oveq12d 6928 . . . . . . . 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 4907 . . . . . 6 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5352ralrimiva 3175 . . . . 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 3175 . . 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 787 . . 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 3161 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6160ad2antll 720 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
62 fveq2 6437 . . . . . . 7 (𝑥 = 𝑎 → (𝑁𝑥) = (𝑁𝑎))
6362eqeq1d 2827 . . . . . 6 (𝑥 = 𝑎 → ((𝑁𝑥) = 0 ↔ (𝑁𝑎) = 0))
64 eqeq1 2829 . . . . . 6 (𝑥 = 𝑎 → (𝑥 = 0𝑎 = 0 ))
6563, 64bibi12d 337 . . . . 5 (𝑥 = 𝑎 → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
6665rspccva 3525 . . . 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 3161 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
7069ad2antll 720 . . . 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 6933 . . . . . . 7 (𝑥 = 𝑎 → (𝑁‘(𝑥 𝑦)) = (𝑁‘(𝑎 𝑦)))
7262oveq1d 6925 . . . . . . 7 (𝑥 = 𝑎 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑦)))
7371, 72breq12d 4888 . . . . . 6 (𝑥 = 𝑎 → ((𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
74 oveq2 6918 . . . . . . . 8 (𝑦 = 𝑏 → (𝑎 𝑦) = (𝑎 𝑏))
7574fveq2d 6441 . . . . . . 7 (𝑦 = 𝑏 → (𝑁‘(𝑎 𝑦)) = (𝑁‘(𝑎 𝑏)))
76 fveq2 6437 . . . . . . . 8 (𝑦 = 𝑏 → (𝑁𝑦) = (𝑁𝑏))
7776oveq2d 6926 . . . . . . 7 (𝑦 = 𝑏 → ((𝑁𝑎) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑏)))
7875, 77breq12d 4888 . . . . . 6 (𝑦 = 𝑏 → ((𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏))))
7973, 78rspc2va 3540 . . . . 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 22834 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑇 ∈ NrmGrp)
8356, 82impbida 835 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 1656  wcel 2164  wral 3117  Vcvv 3414   class class class wbr 4875  wf 6123  cfv 6127  (class class class)co 6910  cr 10258  0cc0 10259   + caddc 10262  cle 10399  Basecbs 16229  +gcplusg 16312  distcds 16321  0gc0g 16460  Grpcgrp 17783  -gcsg 17785  Metcmet 20099  normcnm 22758  NrmGrpcngp 22759   toNrmGrp ctng 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-sup 8623  df-inf 8624  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-uz 11976  df-q 12079  df-rp 12120  df-xneg 12239  df-xadd 12240  df-xmul 12241  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-plusg 16325  df-tset 16331  df-ds 16334  df-rest 16443  df-topn 16444  df-0g 16462  df-topgen 16464  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-sbg 17788  df-psmet 20105  df-xmet 20106  df-met 20107  df-bl 20108  df-mopn 20109  df-top 21076  df-topon 21093  df-topsp 21115  df-bases 21128  df-xms 22502  df-ms 22503  df-nm 22764  df-ngp 22765  df-tng 22766
This theorem is referenced by: (None)
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