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Theorem tngngp 22505
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 2651 . . . . 5 (dist‘𝑇) = (dist‘𝑇)
41, 2, 3tngngp2 22503 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
54simprbda 652 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6 simplr 807 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑇 ∈ NrmGrp)
7 simpr 476 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥𝑋)
8 fvex 6239 . . . . . . . . . . . 12 (Base‘𝐺) ∈ V
92, 8eqeltri 2726 . . . . . . . . . . 11 𝑋 ∈ V
10 reex 10065 . . . . . . . . . . 11 ℝ ∈ V
11 fex2 7163 . . . . . . . . . . 11 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
129, 10, 11mp3an23 1456 . . . . . . . . . 10 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
1312ad2antrr 762 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 ∈ V)
141, 2tngbas 22492 . . . . . . . . 9 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
1513, 14syl 17 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑋 = (Base‘𝑇))
167, 15eleqtrd 2732 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑥 ∈ (Base‘𝑇))
17 eqid 2651 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
18 eqid 2651 . . . . . . . 8 (norm‘𝑇) = (norm‘𝑇)
19 eqid 2651 . . . . . . . 8 (0g𝑇) = (0g𝑇)
2017, 18, 19nmeq0 22469 . . . . . . 7 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
216, 16, 20syl2anc 694 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
225adantr 480 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
23 simpll 805 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁:𝑋⟶ℝ)
241, 2, 10tngnm 22502 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
2522, 23, 24syl2anc 694 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 𝑁 = (norm‘𝑇))
2625fveq1d 6231 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
2726eqeq1d 2653 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
28 tngngp.z . . . . . . . . 9 0 = (0g𝐺)
291, 28tng0 22494 . . . . . . . 8 (𝑁 ∈ V → 0 = (0g𝑇))
3013, 29syl 17 . . . . . . 7 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → 0 = (0g𝑇))
3130eqeq2d 2661 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 = 0𝑥 = (0g𝑇)))
3221, 27, 313bitr4d 300 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
33 simpllr 815 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑇 ∈ NrmGrp)
3416adantr 480 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥 ∈ (Base‘𝑇))
3515eleq2d 2716 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑦𝑋𝑦 ∈ (Base‘𝑇)))
3635biimpa 500 . . . . . . . 8 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦 ∈ (Base‘𝑇))
37 eqid 2651 . . . . . . . . 9 (-g𝑇) = (-g𝑇)
3817, 18, 37nmmtri 22473 . . . . . . . 8 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
3933, 34, 36, 38syl3anc 1366 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
40 tngngp.m . . . . . . . . . . 11 = (-g𝐺)
412, 15syl5eqr 2699 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (Base‘𝐺) = (Base‘𝑇))
42 eqid 2651 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
431, 42tngplusg 22493 . . . . . . . . . . . . 13 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
4413, 43syl 17 . . . . . . . . . . . 12 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (+g𝐺) = (+g𝑇))
4541, 44grpsubpropd 17567 . . . . . . . . . . 11 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (-g𝐺) = (-g𝑇))
4640, 45syl5eq 2697 . . . . . . . . . 10 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → = (-g𝑇))
4746oveqd 6707 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑥 𝑦) = (𝑥(-g𝑇)𝑦))
4825, 47fveq12d 6235 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
4948adantr 480 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) = ((norm‘𝑇)‘(𝑥(-g𝑇)𝑦)))
5025fveq1d 6231 . . . . . . . . 9 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (𝑁𝑦) = ((norm‘𝑇)‘𝑦))
5126, 50oveq12d 6708 . . . . . . . 8 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5251adantr 480 . . . . . . 7 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
5339, 49, 523brtr4d 4717 . . . . . 6 ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5453ralrimiva 2995 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
5532, 54jca 553 . . . 4 (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥𝑋) → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
5655ralrimiva 2995 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
575, 56jca 553 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
58 simprl 809 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝐺 ∈ Grp)
59 simpl 472 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑁:𝑋⟶ℝ)
60 simpl 472 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6160ralimi 2981 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
6261ad2antll 765 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))
63 fveq2 6229 . . . . . . 7 (𝑥 = 𝑎 → (𝑁𝑥) = (𝑁𝑎))
6463eqeq1d 2653 . . . . . 6 (𝑥 = 𝑎 → ((𝑁𝑥) = 0 ↔ (𝑁𝑎) = 0))
65 eqeq1 2655 . . . . . 6 (𝑥 = 𝑎 → (𝑥 = 0𝑎 = 0 ))
6664, 65bibi12d 334 . . . . 5 (𝑥 = 𝑎 → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
6766rspccva 3339 . . . 4 ((∀𝑥𝑋 ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
6862, 67sylan 487 . . 3 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
69 simpr 476 . . . . . 6 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
7069ralimi 2981 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
7170ad2antll 765 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
72 oveq1 6697 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥 𝑦) = (𝑎 𝑦))
7372fveq2d 6233 . . . . . . 7 (𝑥 = 𝑎 → (𝑁‘(𝑥 𝑦)) = (𝑁‘(𝑎 𝑦)))
7463oveq1d 6705 . . . . . . 7 (𝑥 = 𝑎 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑦)))
7573, 74breq12d 4698 . . . . . 6 (𝑥 = 𝑎 → ((𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
76 oveq2 6698 . . . . . . . 8 (𝑦 = 𝑏 → (𝑎 𝑦) = (𝑎 𝑏))
7776fveq2d 6233 . . . . . . 7 (𝑦 = 𝑏 → (𝑁‘(𝑎 𝑦)) = (𝑁‘(𝑎 𝑏)))
78 fveq2 6229 . . . . . . . 8 (𝑦 = 𝑏 → (𝑁𝑦) = (𝑁𝑏))
7978oveq2d 6706 . . . . . . 7 (𝑦 = 𝑏 → ((𝑁𝑎) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑏)))
8077, 79breq12d 4698 . . . . . 6 (𝑦 = 𝑏 → ((𝑁‘(𝑎 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏))))
8175, 80rspc2va 3354 . . . . 5 (((𝑎𝑋𝑏𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8281ancoms 468 . . . 4 ((∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
8371, 82sylan 487 . . 3 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
841, 2, 40, 28, 58, 59, 68, 83tngngpd 22504 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑇 ∈ NrmGrp)
8557, 84impbida 895 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974   + caddc 9977  cle 10113  Basecbs 15904  +gcplusg 15988  distcds 15997  0gc0g 16147  Grpcgrp 17469  -gcsg 17471  Metcme 19780  normcnm 22428  NrmGrpcngp 22429   toNrmGrp ctng 22430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-tset 16007  df-ds 16011  df-rest 16130  df-topn 16131  df-0g 16149  df-topgen 16151  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-xms 22172  df-ms 22173  df-nm 22434  df-ngp 22435  df-tng 22436
This theorem is referenced by: (None)
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