Theorem tngngp3 24881

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 6896	. . . 4 ⊢ 𝑋 ∈ V
3		fex 7228	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
4	2, 3	mpan2 704	. . 3 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
5		tngngp3.t	. . . . . . 7 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
6	5	tnggrpr 24880	. . . . . 6 ⊢ ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
7		simp2 1155	. . . . . . . 8 ⊢ (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝐺 ∈ Grp)
8		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑇) = (Base‘𝑇)
9		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑇) = (norm‘𝑇)
10		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑇) = (0g‘𝑇)
11	8, 9, 10	nmeq0 24843	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇)))
12		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (invg‘𝑇) = (invg‘𝑇)
13	8, 9, 12	nminv 24846	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((invg‘𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥))
14		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	8, 9, 14	nmtri 24851	. . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 487	. . . . . . . . . 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 24866	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
23		tngngp3.p	. . . . . . . . . . . . . . 15 ⊢ + = (+g‘𝐺)
24	5, 23	tngplusg 24867	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → + = (+g‘𝑇))
25		tngngp3.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (invg‘𝐺)
26		eqidd 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝐺))
27		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	5, 27	tngbas 24866	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
29		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝐺) = (+g‘𝐺)
30	5, 29	tngplusg 24867	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ V → (+g‘𝐺) = (+g‘𝑇))
31	30	oveqd 7433	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ V → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
32	31	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ V ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
33	26, 28, 32	grpinvpropd 19137	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ V → (invg‘𝐺) = (invg‘𝑇))
34	25, 33	eqtrid 2809	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 11216	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
39	5, 1, 38	tngnm 24876	. . . . . . . . . . . 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 24868	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ V → 0 = (0g‘𝑇))
43	42	adantr 486	. . . . . . . . . . . 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 6884	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
50	49	eqeq1d 2764	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = (Base‘𝑇) ∧ + = (+g‘𝑇) ∧ 𝐼 = (invg‘𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g‘𝑇)) → (𝐼‘𝑥) = ((invg‘𝑇)‘𝑥))
57	48, 56	fveq12d 6889	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (norm‘𝑇) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
64		fveq1 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (norm‘𝑇) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
65	63, 64	oveq12d 7434	. . . . . . . . . . . . . . 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 5120	. . . . . . . . . . . . 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 3336	. . . . . . . . . . . 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 3336	. . . . . . . . . 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 521	. . . . . . 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 419	. . . 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 2762	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
80		simpl 488	. . . . 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 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))))) → 𝐺 ∈ Grp)
82		simpl 488	. . . 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 6882	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
84	83	eqeq1d 2764	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
85		eqeq1 2766	. . . . . . . . . . . 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 6882	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝐼‘𝑥) = (𝐼‘𝑎))
88	87	fveq2d 6886	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝐼‘𝑥)) = (𝑁‘(𝐼‘𝑎)))
89	88, 83	eqeq12d 2778	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ (𝑁‘(𝐼‘𝑎)) = (𝑁‘𝑎)))
90		fvoveq1 7439	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 + 𝑦)) = (𝑁‘(𝑎 + 𝑦)))
91	83	oveq1d 7431	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
92	90, 91	breq12d 5120	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
93	92	ralbidv 3187	. . . . . . . . . . 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 3578	. . . . . . . . 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 418	. . . . . . 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 487	. . . . . 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 487	. . . . 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 412	. . . 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 19108	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(-g‘𝐺)𝑏) = (𝑎 + (𝐼‘𝑏)))
103	102	adantl 487	. . . . . 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 6886	. . . . 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 3101	. . . . . . . . 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 490	. . . . . . . . . . . . . . . 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 3101	. . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑎 + 𝑦) = (𝑎 + (𝐼‘𝑏)))
110	109	fveq2d 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑁‘(𝑎 + 𝑦)) = (𝑁‘(𝑎 + (𝐼‘𝑏))))
111		fveq2 6882	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼‘𝑏) → (𝑁‘𝑦) = (𝑁‘(𝐼‘𝑏)))
112	111	oveq2d 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼‘𝑏) → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏))))
113	110, 112	breq12d 5120	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐼‘𝑏) → ((𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 + (𝐼‘𝑏))) ≤ ((𝑁‘𝑎) + (𝑁‘(𝐼‘𝑏)))))
114	92, 113	rspc2v 3590	. . . . . . . . . . . . . . . . 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 19110	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋) → (𝐼‘𝑏) ∈ 𝑋)
116	115	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ Grp → (𝑏 ∈ 𝑋 → (𝐼‘𝑏) ∈ 𝑋))
117	116	anim2d 624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ Grp → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎 ∈ 𝑋 ∧ (𝐼‘𝑏) ∈ 𝑋)))
118	117	imp 412	. . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥))
125	124	ralimi 3101	. . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (𝐼‘𝑥) = (𝐼‘𝑏))
127	126	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → (𝑁‘(𝐼‘𝑥)) = (𝑁‘(𝐼‘𝑏)))
128		fveq2 6882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → (𝑁‘𝑥) = (𝑁‘𝑏))
129	127, 128	eqeq12d 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → ((𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ↔ (𝑁‘(𝐼‘𝑏)) = (𝑁‘𝑏)))
130	129	rspccva 3578	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ 𝑏 ∈ 𝑋) → (𝑁‘(𝐼‘𝑏)) = (𝑁‘𝑏))
131	130	eqcomd 2768	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑥 ∈ 𝑋 (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ 𝑏 ∈ 𝑋) → (𝑁‘𝑏) = (𝑁‘(𝐼‘𝑏)))
132	131	ex 418	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . 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 5137	. . . . . . . . . . 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 418	. . . . . . . . . 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 418	. . . . . . . . 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 413	. . . . . . 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 487	. . . . . 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 412	. . . . 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 5131	. . . 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 24878	. . 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 418	. 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 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   class class class wbr 5107  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  ℝcr 11124  0cc0 11125   + caddc 11128   ≤ cle 11269  Basecbs 17303  +_gcplusg 17344  0_gc0g 17526  Grpcgrp 19056  inv_gcminusg 19057  -_gcsg 19058  normcnm 24801  NrmGrpcngp 24802   toNrmGrp ctng 24803

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202  ax-pre-sup 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-inf 9416  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-div 11897  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-dec 12738  df-uz 12889  df-q 12999  df-rp 13043  df-xneg 13163  df-xadd 13164  df-xmul 13165  df-sets 17258  df-slot 17276  df-ndx 17288  df-base 17304  df-plusg 17357  df-tset 17363  df-ds 17366  df-rest 17509  df-topn 17510  df-0g 17528  df-topgen 17530  df-mgm 18732  df-sgrp 18821  df-mnd 18837  df-grp 19059  df-minusg 19060  df-sbg 19061  df-psmet 21576  df-xmet 21577  df-met 21578  df-bl 21579  df-mopn 21580  df-top 23118  df-topon 23135  df-topsp 23157  df-bases 23170  df-xms 24545  df-ms 24546  df-nm 24807  df-ngp 24808  df-tng 24809

This theorem is used by: (None)