Theorem tngngp3 24822

Description: Alternate definition of a normed group (i.e., a group equipped with a norm) without using the properties of a metric space. This corresponds to the definition in N. H. Bingham, A. J. Ostaszewski: "Normed versus topological groups: dichotomy and duality", 2010, Dissertationes Mathematicae 472, pp. 1-138 and E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006. (Contributed by AV, 16-Oct-2021.)

Hypotheses

Ref	Expression
tngngp3.t	⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
tngngp3.x	⊢ 𝑋 = (Base‘𝐺)
tngngp3.z	⊢ 0 = (0g‘𝐺)
tngngp3.p	⊢ + = (+g‘𝐺)
tngngp3.i	⊢ 𝐼 = (invg‘𝐺)

Assertion

Ref	Expression
tngngp3	⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑁,𝑦   𝑥,𝑇,𝑦   𝑥,𝑋,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦   𝑥, 0 ,𝑦

Proof of Theorem tngngp3

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tngngp3.x	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
2	1	fvexi 6895	. . . 4 ⊢ 𝑋 ∈ V
3		fex 7224	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
4	2, 3	mpan2 703	. . 3 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
5		tngngp3.t	. . . . . . 7 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
6	5	tnggrpr 24821	. . . . . 6 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
7		simp2 1155	. . . . . . . 8 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝐺 ∈ Grp)
8		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑇) = (Base‘𝑇)
9		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑇) = (norm‘𝑇)
10		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑇) = (0g‘𝑇)
11	8, 9, 10	nmeq0 24784	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)))
12		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (invg‘𝑇) = (invg‘𝑇)
13	8, 9, 12	nminv 24787	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥))
14		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	8, 9, 14	nmtri 24792	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
16	15	3expa 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
17	16	ralrimiva 3157	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
18	11, 13, 17	3jca 1146	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
19	18	ralrimiva 3157	. . . . . . . . . . 11 ⊢ (𝑇 ∈ NrmGrp → ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
20	19	adantl 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
21	20	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
22	5, 1	tngbas 24807	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
23		tngngp3.p	. . . . . . . . . . . . . . 15 ⊢ + = (+g‘𝐺)
24	5, 23	tngplusg 24808	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → + = (+g‘𝑇))
25		tngngp3.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (invg‘𝐺)
26		eqidd 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝐺))
27		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	5, 27	tngbas 24807	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
29		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝐺) = (+g‘𝐺)
30	5, 29	tngplusg 24808	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ V → (+g‘𝐺) = (+g‘𝑇))
31	30	oveqd 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ V → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
32	31	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ V ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
33	26, 28, 32	grpinvpropd 19085	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ V → (invg‘𝐺) = (invg‘𝑇))
34	25, 33	eqtrid 2810	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → 𝐼 = (invg‘𝑇))
35	22, 24, 34	3jca 1146	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ V → (𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)))
36	35	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)))
37	36	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)))
38		reex 11195	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
39	5, 1, 38	tngnm 24817	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
40	39	3adant1 1148	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
41		tngngp3.z	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
42	5, 41	tng0 24809	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ V → 0 = (0g‘𝑇))
43	42	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 0 = (0g‘𝑇))
44	43	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 0 = (0g‘𝑇))
45	37, 40, 44	3jca 1146	. . . . . . . . . 10 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)))
46		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) → 𝑋 = (Base‘𝑇))
47	46	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 𝑋 = (Base‘𝑇))
48		simp2 1155	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 𝑁 = (norm‘𝑇))
49	48	fveq1d 6883	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
50	49	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → ((𝑁‘𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
51		simp3 1156	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 0 = (0g‘𝑇))
52	51	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑥 = 0 ↔ 𝑥 = (0g‘𝑇)))
53	50, 52	bibi12d 348	. . . . . . . . . . . 12 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ↔ (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇))))
54		simp3 1156	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) → 𝐼 = (invg‘𝑇))
55	54	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 𝐼 = (invg‘𝑇))
56	55	fveq1d 6883	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝐼‘𝑥) = ((invg‘𝑇)‘𝑥))
57	48, 56	fveq12d 6888	. . . . . . . . . . . . 13 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑁‘(𝐼‘𝑥)) = ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)))
58	57, 49	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥)))
59		simp2 1155	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) → + = (+g‘𝑇))
60	59	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → + = (+g‘𝑇))
61	60	oveqd 7427	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑥 + 𝑦) = (𝑥(+g‘𝑇)𝑦))
62	48, 61	fveq12d 6888	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑁‘(𝑥 + 𝑦)) = ((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)))
63		fveq1 6880	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (norm‘𝑇) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
64		fveq1 6880	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (norm‘𝑇) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
65	63, 64	oveq12d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (norm‘𝑇) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
66	65	3ad2ant2 1152	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
67	62, 66	breq12d 5122	. . . . . . . . . . . . 13 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → ((𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ ((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
68	47, 67	raleqbidv 3338	. . . . . . . . . . . 12 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
69	53, 58, 68	3anbi123d 1464	. . . . . . . . . . 11 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ↔ ((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))))
70	47, 69	raleqbidv 3338	. . . . . . . . . 10 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))))
71	45, 70	syl 18	. . . . . . . . 9 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)) ∧ ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))))
72	21, 71	mpbird 260	. . . . . . . 8 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
73	7, 72	jca 520	. . . . . . 7 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
74	73	3exp 1137	. . . . . 6 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp → (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))))
75	6, 74	mpd 16	. . . . 5 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
76	75	expcom 418	. . . 4 ⊢ (𝑇 ∈ NrmGrp → (𝑁 ∈ V → (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))))
77	76	com13 89	. . 3 ⊢ (𝑁:𝑋⟶ℝ → (𝑁 ∈ V → (𝑇 ∈ NrmGrp → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))))
78	4, 77	mpd 16	. 2 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
79		eqid 2763	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
80		simpl 487	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → 𝐺 ∈ Grp)
81	80	adantl 486	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝐺 ∈ Grp)
82		simpl 487	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝑁:𝑋⟶ℝ)
83		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
84	83	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
85		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 = 0 ↔ 𝑎 = 0 ))
86	84, 85	bibi12d 348	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )))
87		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝐼‘𝑥) = (𝐼‘𝑎))
88	87	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝐼‘𝑥)) = (𝑁‘(𝐼‘𝑎)))
89	88, 83	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ (𝑁‘(𝐼‘𝑎)) = (𝑁‘𝑎)))
90		fvoveq1 7433	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 + 𝑦)) = (𝑁‘(𝑎 + 𝑦)))
91	83	oveq1d 7425	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
92	90, 91	breq12d 5122	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
93	92	ralbidv 3188	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
94	86, 89, 93	3anbi123d 1464	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ↔ (((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ) ∧ (𝑁‘(𝐼‘𝑎)) = (𝑁‘𝑎) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)))))
95	94	rspccva 3580	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝑎 ∈ 𝑋) → (((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ) ∧ (𝑁‘(𝐼‘𝑎)) = (𝑁‘𝑎) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
96		simp1 1154	. . . . . . . . 9 ⊢ ((((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ) ∧ (𝑁‘(𝐼‘𝑎)) = (𝑁‘𝑎) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
97	95, 96	syl 18	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
98	97	ex 417	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑎 ∈ 𝑋 → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )))
99	98	adantl 486	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → (𝑎 ∈ 𝑋 → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )))
100	99	adantl 486	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → (𝑎 ∈ 𝑋 → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )))
101	100	imp 411	. . . 4 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
102	1, 23, 25, 79	grpsubval 19056	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(-g‘𝐺)𝑏) = (𝑎 + (𝐼‘𝑏)))
103	102	adantl 486	. . . . . 6 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(-g‘𝐺)𝑏) = (𝑎 + (𝐼‘𝑏)))
104	103	fveq2d 6885	. . . . 5 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎(-g‘𝐺)𝑏)) = (𝑁‘(𝑎 + (𝐼‘𝑏))))
105		3simpc 1168	. . . . . . . . . 10 ⊢ ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
106	105	ralimi 3102	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
107		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
108	107	ralimi 3102	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
109		oveq2 7418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑎 + 𝑦) = (𝑎 + (𝐼‘𝑏)))
110	109	fveq2d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑁‘(𝑎 + 𝑦)) = (𝑁‘(𝑎 + (𝐼‘𝑏))))
111		fveq2 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑁‘𝑦) = (𝑁‘(𝐼‘𝑏)))
112	111	oveq2d 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼‘𝑏) → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))
113	110, 112	breq12d 5122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐼‘𝑏) → ((𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏)))))
114	92, 113	rspc2v 3592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝑋 ∧ (𝐼‘𝑏) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏)))))
115	1, 25	grpinvcl 19058	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋) → (𝐼‘𝑏) ∈ 𝑋)
116	115	ex 417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ Grp → (𝑏 ∈ 𝑋 → (𝐼‘𝑏) ∈ 𝑋))
117	116	anim2d 623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎 ∈ 𝑋 ∧ (𝐼‘𝑏) ∈ 𝑋)))
118	117	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 ∈ 𝑋 ∧ (𝐼‘𝑏) ∈ 𝑋))
119	114, 118	syl11 34	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) → ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏)))))
120	119	expd 420	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) → (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))))
121	108, 120	syl 18	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))))
122	121	imp 411	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏)))))
123	122	imp 411	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))
124		simpl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥))
125	124	ralimi 3102	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥))
126		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (𝐼‘𝑥) = (𝐼‘𝑏))
127	126	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → (𝑁‘(𝐼‘𝑥)) = (𝑁‘(𝐼‘𝑏)))
128		fveq2 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → (𝑁‘𝑥) = (𝑁‘𝑏))
129	127, 128	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ (𝑁‘(𝐼‘𝑏)) = (𝑁‘𝑏)))
130	129	rspccva 3580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝐼‘𝑏)) = (𝑁‘𝑏))
131	130	eqcomd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ 𝑏 ∈ 𝑋) → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏)))
132	131	ex 417	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) → (𝑏 ∈ 𝑋 → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏))))
133	125, 132	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑏 ∈ 𝑋 → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏))))
134	133	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) → (𝑏 ∈ 𝑋 → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏))))
135	134	adantld 495	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏))))
136	135	imp 411	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏)))
137	136	oveq2d 7426	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑁‘𝑎) + (𝑁‘𝑏)) = ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))
138	123, 137	breqtrrd 5139	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))
139	138	ex 417	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ 𝐺 ∈ Grp) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
140	139	ex 417	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))))
141	106, 140	syl 18	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))))
142	141	impcom 412	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
143	142	adantl 486	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
144	143	imp 411	. . . . 5 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))
145	104, 144	eqbrtrd 5133	. . . 4 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎(-g‘𝐺)𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))
146	5, 1, 79, 41, 81, 82, 101, 145	tngngpd 24819	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝑇 ∈ NrmGrp)
147	146	ex 417	. 2 ⊢ (𝑁:𝑋⟶ℝ → ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → 𝑇 ∈ NrmGrp))
148	78, 147	impbid 215	1 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℝcr 11103  0cc0 11104   + caddc 11107   ≤ cle 11248  Basecbs 17273  +_gcplusg 17314  0_gc0g 17496  Grpcgrp 19004  inv_gcminusg 19005  -_gcsg 19006  normcnm 24742  NrmGrpcngp 24743   toNrmGrp ctng 24744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181  ax-pre-sup 11182

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-inf 9399  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-div 11876  df-nn 12238  df-2 12307  df-3 12308  df-4 12309  df-5 12310  df-6 12311  df-7 12312  df-8 12313  df-9 12314  df-n0 12509  df-z 12596  df-dec 12716  df-uz 12867  df-q 12977  df-rp 13021  df-xneg 13141  df-xadd 13142  df-xmul 13143  df-sets 17228  df-slot 17246  df-ndx 17258  df-base 17274  df-plusg 17327  df-tset 17333  df-ds 17336  df-rest 17479  df-topn 17480  df-0g 17498  df-topgen 17500  df-mgm 18702  df-sgrp 18781  df-mnd 18797  df-grp 19007  df-minusg 19008  df-sbg 19009  df-psmet 21523  df-xmet 21524  df-met 21525  df-bl 21526  df-mopn 21527  df-top 23060  df-topon 23077  df-topsp 23099  df-bases 23112  df-xms 24486  df-ms 24487  df-nm 24748  df-ngp 24749  df-tng 24750

This theorem is used by: (None)