Theorem tngngp3 24643

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 6845	. . . 4 ⊢ 𝑋 ∈ V
3		fex 7174	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
4	2, 3	mpan2 698	. . 3 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
5		tngngp3.t	. . . . . . 7 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
6	5	tnggrpr 24642	. . . . . 6 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
7		simp2 1144	. . . . . . . 8 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝐺 ∈ Grp)
8		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑇) = (Base‘𝑇)
9		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑇) = (norm‘𝑇)
10		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑇) = (0g‘𝑇)
11	8, 9, 10	nmeq0 24605	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)))
12		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (invg‘𝑇) = (invg‘𝑇)
13	8, 9, 12	nminv 24608	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥))
14		eqid 2741	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	8, 9, 14	nmtri 24613	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
16	15	3expa 1125	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(+g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
17	16	ralrimiva 3133	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . 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 3133	. . . . . . . . . . 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 483	. . . . . . . . . 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 1140	. . . . . . . . 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 24628	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
23		tngngp3.p	. . . . . . . . . . . . . . 15 ⊢ + = (+g‘𝐺)
24	5, 23	tngplusg 24629	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → + = (+g‘𝑇))
25		tngngp3.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (invg‘𝐺)
26		eqidd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝐺))
27		eqid 2741	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	5, 27	tngbas 24628	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
29		eqid 2741	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝐺) = (+g‘𝐺)
30	5, 29	tngplusg 24629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ V → (+g‘𝐺) = (+g‘𝑇))
31	30	oveqd 7377	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ V → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
32	31	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ V ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
33	26, 28, 32	grpinvpropd 18986	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ V → (invg‘𝐺) = (invg‘𝑇))
34	25, 33	eqtrid 2788	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → 𝐼 = (invg‘𝑇))
35	22, 24, 34	3jca 1135	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ V → (𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)))
36	35	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)))
37	36	3ad2ant1 1140	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)))
38		reex 11124	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
39	5, 1, 38	tngnm 24638	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
40	39	3adant1 1137	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
41		tngngp3.z	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
42	5, 41	tng0 24630	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ V → 0 = (0g‘𝑇))
43	42	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 0 = (0g‘𝑇))
44	43	3ad2ant1 1140	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 0 = (0g‘𝑇))
45	37, 40, 44	3jca 1135	. . . . . . . . . 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 1143	. . . . . . . . . . . 12 ⊢ ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) → 𝑋 = (Base‘𝑇))
47	46	3ad2ant1 1140	. . . . . . . . . . 11 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 𝑋 = (Base‘𝑇))
48		simp2 1144	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 𝑁 = (norm‘𝑇))
49	48	fveq1d 6833	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
50	49	eqeq1d 2743	. . . . . . . . . . . . 13 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → ((𝑁‘𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
51		simp3 1145	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 0 = (0g‘𝑇))
52	51	eqeq2d 2752	. . . . . . . . . . . . 13 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑥 = 0 ↔ 𝑥 = (0g‘𝑇)))
53	50, 52	bibi12d 347	. . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) → 𝐼 = (invg‘𝑇))
55	54	3ad2ant1 1140	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → 𝐼 = (invg‘𝑇))
56	55	fveq1d 6833	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝐼‘𝑥) = ((invg‘𝑇)‘𝑥))
57	48, 56	fveq12d 6838	. . . . . . . . . . . . 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 2757	. . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) → + = (+g‘𝑇))
60	59	3ad2ant1 1140	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → + = (+g‘𝑇))
61	60	oveqd 7377	. . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (norm‘𝑇) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
64		fveq1 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (norm‘𝑇) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
65	63, 64	oveq12d 7378	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (norm‘𝑇) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
66	65	3ad2ant2 1141	. . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . 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 3315	. . . . . . . . . . . 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 1445	. . . . . . . . . . 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 3315	. . . . . . . . . 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 17	. . . . . . . . 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 259	. . . . . . . 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 517	. . . . . . 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 1126	. . . . . 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 15	. . . . 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 415	. . . 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 88	. . 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 15	. 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 2741	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
80		simpl 484	. . . . 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 483	. . . 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 484	. . . 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 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
84	83	eqeq1d 2743	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
85		eqeq1 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 = 0 ↔ 𝑎 = 0 ))
86	84, 85	bibi12d 347	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )))
87		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝐼‘𝑥) = (𝐼‘𝑎))
88	87	fveq2d 6835	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝐼‘𝑥)) = (𝑁‘(𝐼‘𝑎)))
89	88, 83	eqeq12d 2757	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ (𝑁‘(𝐼‘𝑎)) = (𝑁‘𝑎)))
90		fvoveq1 7383	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 + 𝑦)) = (𝑁‘(𝑎 + 𝑦)))
91	83	oveq1d 7375	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
92	90, 91	breq12d 5088	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
93	92	ralbidv 3164	. . . . . . . . . . 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 1445	. . . . . . . . . 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 3561	. . . . . . . . 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 1143	. . . . . . . . 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 17	. . . . . . . 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 414	. . . . . . 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 483	. . . . . 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 483	. . . . 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 408	. . . 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 18956	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(-g‘𝐺)𝑏) = (𝑎 + (𝐼‘𝑏)))
103	102	adantl 483	. . . . . 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 6835	. . . . 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 1157	. . . . . . . . . 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 3078	. . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 3078	. . . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑎 + 𝑦) = (𝑎 + (𝐼‘𝑏)))
110	109	fveq2d 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑁‘(𝑎 + 𝑦)) = (𝑁‘(𝑎 + (𝐼‘𝑏))))
111		fveq2 6831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑁‘𝑦) = (𝑁‘(𝐼‘𝑏)))
112	111	oveq2d 7376	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼‘𝑏) → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))
113	110, 112	breq12d 5088	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐼‘𝑏) → ((𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏)))))
114	92, 113	rspc2v 3573	. . . . . . . . . . . . . . . . 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 18958	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋) → (𝐼‘𝑏) ∈ 𝑋)
116	115	ex 414	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ Grp → (𝑏 ∈ 𝑋 → (𝐼‘𝑏) ∈ 𝑋))
117	116	anim2d 619	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎 ∈ 𝑋 ∧ (𝐼‘𝑏) ∈ 𝑋)))
118	117	imp 408	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 ∈ 𝑋 ∧ (𝐼‘𝑏) ∈ 𝑋))
119	114, 118	syl11 33	. . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 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 408	. . . . . . . . . . . . 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 408	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥))
125	124	ralimi 3078	. . . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (𝐼‘𝑥) = (𝐼‘𝑏))
127	126	fveq2d 6835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → (𝑁‘(𝐼‘𝑥)) = (𝑁‘(𝐼‘𝑏)))
128		fveq2 6831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → (𝑁‘𝑥) = (𝑁‘𝑏))
129	127, 128	eqeq12d 2757	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ (𝑁‘(𝐼‘𝑏)) = (𝑁‘𝑏)))
130	129	rspccva 3561	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝐼‘𝑏)) = (𝑁‘𝑏))
131	130	eqcomd 2747	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ 𝑏 ∈ 𝑋) → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏)))
132	131	ex 414	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) → (𝑏 ∈ 𝑋 → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏))))
133	125, 132	syl 17	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . 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 408	. . . . . . . . . . . . 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 7376	. . . . . . . . . . . 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 5103	. . . . . . . . . . 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 414	. . . . . . . . . 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 414	. . . . . . . . 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 17	. . . . . . . 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 409	. . . . . . 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 483	. . . . . 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 408	. . . . 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 5097	. . . 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 24640	. . 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 414	. 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 214	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))))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   class class class wbr 5075  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  ℝcr 11032  0cc0 11033   + caddc 11036   ≤ cle 11175  Basecbs 17174  +_gcplusg 17215  0_gc0g 17397  Grpcgrp 18904  inv_gcminusg 18905  -_gcsg 18906  normcnm 24563  NrmGrpcngp 24564   toNrmGrp ctng 24565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-plusg 17228  df-tset 17234  df-ds 17237  df-rest 17380  df-topn 17381  df-0g 17399  df-topgen 17401  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-sbg 18909  df-psmet 21343  df-xmet 21344  df-met 21345  df-bl 21346  df-mopn 21347  df-top 22881  df-topon 22898  df-topsp 22920  df-bases 22933  df-xms 24307  df-ms 24308  df-nm 24569  df-ngp 24570  df-tng 24571

This theorem is referenced by: (None)