MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tngngp Structured version   Visualization version   GIF version

Theorem tngngp 23190
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 2818 . . . . 5 (dist‘𝑇) = (dist‘𝑇)
41, 2, 3tngngp2 23188 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
54simprbda 499 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6 simplr 765 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑇 ∈ NrmGrp)
7 simpr 485 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥𝑋)
82fvexi 6677 . . . . . . . . . . 11 𝑋 ∈ V
9 reex 10616 . . . . . . . . . . 11 ℝ ∈ V
10 fex2 7627 . . . . . . . . . . 11 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
118, 9, 10mp3an23 1444 . . . . . . . . . 10 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
1211ad2antrr 722 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 ∈ V)
131, 2tngbas 23177 . . . . . . . . 9 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
1412, 13syl 17 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑋 = (Base‘𝑇))
157, 14eleqtrd 2912 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥 ∈ (Base‘𝑇))
16 eqid 2818 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
17 eqid 2818 . . . . . . . 8 (norm‘𝑇) = (norm‘𝑇)
18 eqid 2818 . . . . . . . 8 (0g𝑇) = (0g𝑇)
1916, 17, 18nmeq0 23154 . . . . . . 7 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
206, 15, 19syl2anc 584 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
215adantr 481 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
22 simpll 763 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁:𝑋⟶ℝ)
231, 2, 9tngnm 23187 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
2421, 22, 23syl2anc 584 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 = (norm‘𝑇))
2524fveq1d 6665 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
2625eqeq1d 2820 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
27 tngngp.z . . . . . . . . 9 0 = (0g𝐺)
281, 27tng0 23179 . . . . . . . 8 (𝑁 ∈ V → 0 = (0g𝑇))
2912, 28syl 17 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 0 = (0g𝑇))
3029eqeq2d 2829 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 = 0𝑥 = (0g𝑇)))
3120, 26, 303bitr4d 312 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
32 simpllr 772 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑇 ∈ NrmGrp)
3315adantr 481 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥 ∈ (Base‘𝑇))
3414eleq2d 2895 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑦𝑋𝑦 ∈ (Base‘𝑇)))
3534biimpa 477 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦 ∈ (Base‘𝑇))
36 eqid 2818 . . . . . . . . 9 (-g𝑇) = (-g𝑇)
3716, 17, 36nmmtri 23158 . . . . . . . 8 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
3832, 33, 35, 37syl3anc 1363 . . . . . . 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 2867 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (Base‘𝐺) = (Base‘𝑇))
41 eqid 2818 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
421, 41tngplusg 23178 . . . . . . . . . . . . 13 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
4312, 42syl 17 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (+g𝐺) = (+g𝑇))
4440, 43grpsubpropd 18142 . . . . . . . . . . 11 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (-g𝐺) = (-g𝑇))
4539, 44syl5eq 2865 . . . . . . . . . 10 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → = (-g𝑇))
4645oveqd 7162 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 𝑦) = (𝑥(-g𝑇)𝑦))
4724, 46fveq12d 6670 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4847adantr 481 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4924fveq1d 6665 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑦) = ((norm‘𝑇)‘𝑦))
5025, 49oveq12d 7163 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5150adantr 481 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5238, 48, 513brtr4d 5089 . . . . . 6 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5352ralrimiva 3179 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5431, 53jca 512 . . . 4 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
5554ralrimiva 3179 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
565, 55jca 512 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
57 simprl 767 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝐺 ∈ Grp)
58 simpl 483 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑁:𝑋⟶ℝ)
59 simpl 483 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6059ralimi 3157 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6160ad2antll 725 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
62 fveq2 6663 . . . . . . 7 (𝑥 = 𝑎 → (𝑁𝑥) = (𝑁𝑎))
6362eqeq1d 2820 . . . . . 6 (𝑥 = 𝑎 → ((𝑁𝑥) = 0 ↔ (𝑁𝑎) = 0))
64 eqeq1 2822 . . . . . 6 (𝑥 = 𝑎 → (𝑥 = 0𝑎 = 0 ))
6563, 64bibi12d 347 . . . . 5 (𝑥 = 𝑎 → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
6665rspccva 3619 . . . 4 ((∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
6761, 66sylan 580 . . 3 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
68 simpr 485 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
6968ralimi 3157 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
7069ad2antll 725 . . . 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 7168 . . . . . . 7 (𝑥 = 𝑎 → (𝑁‘(𝑥 𝑦)) = (𝑁‘(𝑎 𝑦)))
7262oveq1d 7160 . . . . . . 7 (𝑥 = 𝑎 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑦)))
7371, 72breq12d 5070 . . . . . 6 (𝑥 = 𝑎 → ((𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
74 oveq2 7153 . . . . . . . 8 (𝑦 = 𝑏 → (𝑎 𝑦) = (𝑎 𝑏))
7574fveq2d 6667 . . . . . . 7 (𝑦 = 𝑏 → (𝑁‘(𝑎 𝑦)) = (𝑁‘(𝑎 𝑏)))
76 fveq2 6663 . . . . . . . 8 (𝑦 = 𝑏 → (𝑁𝑦) = (𝑁𝑏))
7776oveq2d 7161 . . . . . . 7 (𝑦 = 𝑏 → ((𝑁𝑎) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑏)))
7875, 77breq12d 5070 . . . . . 6 (𝑦 = 𝑏 → ((𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏))))
7973, 78rspc2va 3631 . . . . 5 (((𝑎𝑋𝑏𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8079ancoms 459 . . . 4 ((∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8170, 80sylan 580 . . 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 23189 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑇 ∈ NrmGrp)
8356, 82impbida 797 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525   + caddc 10528  cle 10664  Basecbs 16471  +gcplusg 16553  distcds 16562  0gc0g 16701  Grpcgrp 18041  -gcsg 18043  Metcmet 20459  normcnm 23113  NrmGrpcngp 23114   toNrmGrp ctng 23115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-plusg 16566  df-tset 16572  df-ds 16575  df-rest 16684  df-topn 16685  df-0g 16703  df-topgen 16705  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-xms 22857  df-ms 22858  df-nm 23119  df-ngp 23120  df-tng 23121
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator