Theorem nmhmcn 23723

Description: A linear operator over a normed subcomplex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)

Hypotheses

Ref	Expression
nmhmcn.j	⊢ 𝐽 = (TopOpen‘𝑆)
nmhmcn.k	⊢ 𝐾 = (TopOpen‘𝑇)
nmhmcn.g	⊢ 𝐺 = (Scalar‘𝑆)
nmhmcn.b	⊢ 𝐵 = (Base‘𝐺)

Assertion

Ref	Expression
nmhmcn	⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))

Proof of Theorem nmhmcn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinel1 4171	. . . 4 ⊢ (𝑆 ∈ (NrmMod ∩ ℂMod) → 𝑆 ∈ NrmMod)
2		elinel1 4171	. . . 4 ⊢ (𝑇 ∈ (NrmMod ∩ ℂMod) → 𝑇 ∈ NrmMod)
3		isnmhm 23354	. . . . 5 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
4	3	baib 538	. . . 4 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
5	1, 2, 4	syl2an 597	. . 3 ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
6	5	3adant3 1128	. 2 ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
7		nmhmcn.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑆)
8		nmhmcn.k	. . . . 5 ⊢ 𝐾 = (TopOpen‘𝑇)
9	7, 8	nghmcn 23353	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
10		simpll1 1208	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
11	10	elin1d 4174	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmMod)
12		nlmngp 23285	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
13		ngpms 23208	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
14	11, 12, 13	3syl 18	. . . . . . . 8 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ MetSp)
15		msxms 23063	. . . . . . . 8 ⊢ (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
16		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		eqid 2821	. . . . . . . . 9 ⊢ ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
18	16, 17	xmsxmet 23065	. . . . . . . 8 ⊢ (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
19	14, 15, 18	3syl 18	. . . . . . 7 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
20		simpr 487	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
21		simpll2 1209	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
22	21	elin1d 4174	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmMod)
23		nlmngp 23285	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
24		ngpms 23208	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
25	22, 23, 24	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ MetSp)
26		msxms 23063	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
27		eqid 2821	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
28		eqid 2821	. . . . . . . . . . . . 13 ⊢ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
29	27, 28	xmsxmet 23065	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
30	25, 26, 29	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
31		nlmlmod 23286	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ LMod)
32		eqid 2821	. . . . . . . . . . . . 13 ⊢ (0g‘𝑇) = (0g‘𝑇)
33	27, 32	lmod0vcl 19662	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ LMod → (0g‘𝑇) ∈ (Base‘𝑇))
34	22, 31, 33	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g‘𝑇) ∈ (Base‘𝑇))
35		1rp 12392	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ+
36		rpxr 12397	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*)
37	35, 36	mp1i 13	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 1 ∈ ℝ*)
38		eqid 2821	. . . . . . . . . . . 12 ⊢ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
39	38	blopn 23109	. . . . . . . . . . 11 ⊢ ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (0g‘𝑇) ∈ (Base‘𝑇) ∧ 1 ∈ ℝ*) → ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
40	30, 34, 37, 39	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
41	8, 27, 28	mstopn 23061	. . . . . . . . . . 11 ⊢ (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
42	22, 23, 24, 41	4syl 19	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
43	40, 42	eleqtrrd 2916	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ 𝐾)
44		cnima 21872	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ 𝐾) → (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ 𝐽)
45	20, 43, 44	syl2anc 586	. . . . . . . 8 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ 𝐽)
46	7, 16, 17	mstopn 23061	. . . . . . . . 9 ⊢ (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
47	11, 12, 13, 46	4syl 19	. . . . . . . 8 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
48	45, 47	eleqtrd 2915	. . . . . . 7 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
49		nlmlmod 23286	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod)
50		eqid 2821	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
51	16, 50	lmod0vcl 19662	. . . . . . . . 9 ⊢ (𝑆 ∈ LMod → (0g‘𝑆) ∈ (Base‘𝑆))
52	11, 49, 51	3syl 18	. . . . . . . 8 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g‘𝑆) ∈ (Base‘𝑆))
53		lmghm 19802	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
54	53	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
55	50, 32	ghmid 18363	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
56	54, 55	syl 17	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
57	35	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 1 ∈ ℝ+)
58		blcntr 23022	. . . . . . . . . 10 ⊢ ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (0g‘𝑇) ∈ (Base‘𝑇) ∧ 1 ∈ ℝ+) → (0g‘𝑇) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))
59	30, 34, 57, 58	syl3anc 1367	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g‘𝑇) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))
60	56, 59	eqeltrd 2913	. . . . . . . 8 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹‘(0g‘𝑆)) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))
61	16, 27	lmhmf 19805	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
62	61	ad2antlr 725	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
63		ffn 6513	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
64		elpreima 6827	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → ((0g‘𝑆) ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ((0g‘𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g‘𝑆)) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
65	62, 63, 64	3syl 18	. . . . . . . 8 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((0g‘𝑆) ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ((0g‘𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g‘𝑆)) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
66	52, 60, 65	mpbir2and 711	. . . . . . 7 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g‘𝑆) ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))
67		eqid 2821	. . . . . . . 8 ⊢ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
68	67	mopni2 23102	. . . . . . 7 ⊢ ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) ∧ (0g‘𝑆) ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))) → ∃𝑥 ∈ ℝ+ ((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))
69	19, 48, 66, 68	syl3anc 1367	. . . . . 6 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∃𝑥 ∈ ℝ+ ((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))
70		simpl1 1187	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
71	70	elin1d 4174	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ NrmMod)
72	71, 12	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ NrmGrp)
73	72	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmGrp)
74	73	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ NrmGrp)
75		ngpgrp 23207	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp)
76	74, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ Grp)
77		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
78		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (norm‘𝑆) = (norm‘𝑆)
79		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (dist‘𝑆) = (dist‘𝑆)
80	78, 16, 50, 79, 17	nmval2 23200	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑦) = (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g‘𝑆)))
81	76, 77, 80	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑦) = (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g‘𝑆)))
82	19	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
83	52	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (0g‘𝑆) ∈ (Base‘𝑆))
84		xmetsym 22956	. . . . . . . . . . . . . . 15 ⊢ ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g‘𝑆)) = ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
85	82, 77, 83, 84	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g‘𝑆)) = ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
86	81, 85	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑦) = ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
87	86	breq1d 5075	. . . . . . . . . . . 12 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑆)‘𝑦) < 𝑥 ↔ ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥))
88	87	biimpd 231	. . . . . . . . . . 11 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑆)‘𝑦) < 𝑥 → ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥))
89	62	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
90		elpreima 6827	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
91	89, 63, 90	3syl 18	. . . . . . . . . . . 12 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
92	30	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
93	34	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (0g‘𝑇) ∈ (Base‘𝑇))
94	35, 36	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 1 ∈ ℝ*)
95		elbl 22997	. . . . . . . . . . . . . . 15 ⊢ ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (0g‘𝑇) ∈ (Base‘𝑇) ∧ 1 ∈ ℝ*) → ((𝐹‘𝑦) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ↔ ((𝐹‘𝑦) ∈ (Base‘𝑇) ∧ ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 1)))
96	92, 93, 94, 95	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ↔ ((𝐹‘𝑦) ∈ (Base‘𝑇) ∧ ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 1)))
97		simpl2 1188	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
98	97	elin1d 4174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ NrmMod)
99	98, 23	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ NrmGrp)
100	99	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmGrp)
101	100	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ NrmGrp)
102		simplr 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
103	102	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝐹 ∈ (𝑆 LMHom 𝑇))
104	103, 61	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
105	104	ffvelrnda 6850	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑦) ∈ (Base‘𝑇))
106		eqid 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (norm‘𝑇) = (norm‘𝑇)
107	27, 106	nmcl 23224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑦)) ∈ ℝ)
108	101, 105, 107	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑦)) ∈ ℝ)
109		1re 10640	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
110		ltle 10728	. . . . . . . . . . . . . . . . 17 ⊢ ((((norm‘𝑇)‘(𝐹‘𝑦)) ∈ ℝ ∧ 1 ∈ ℝ) → (((norm‘𝑇)‘(𝐹‘𝑦)) < 1 → ((norm‘𝑇)‘(𝐹‘𝑦)) ≤ 1))
111	108, 109, 110	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑇)‘(𝐹‘𝑦)) < 1 → ((norm‘𝑇)‘(𝐹‘𝑦)) ≤ 1))
112		ngpgrp 23207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
113	101, 112	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ Grp)
114		eqid 2821	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dist‘𝑇) = (dist‘𝑇)
115	106, 27, 32, 114, 28	nmval2 23200	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑦)) = ((𝐹‘𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g‘𝑇)))
116	113, 105, 115	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑦)) = ((𝐹‘𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g‘𝑇)))
117		xmetsym 22956	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (𝐹‘𝑦) ∈ (Base‘𝑇) ∧ (0g‘𝑇) ∈ (Base‘𝑇)) → ((𝐹‘𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g‘𝑇)) = ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)))
118	92, 105, 93, 117	syl3anc 1367	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g‘𝑇)) = ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)))
119	116, 118	eqtrd 2856	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑦)) = ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)))
120	119	breq1d 5075	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑇)‘(𝐹‘𝑦)) < 1 ↔ ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 1))
121		1red 10641	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 1 ∈ ℝ)
122		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ ℝ+)
123	108, 121, 122	lediv1d 12476	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑇)‘(𝐹‘𝑦)) ≤ 1 ↔ (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥)))
124	111, 120, 123	3imtr3d 295	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 1 → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥)))
125	124	adantld 493	. . . . . . . . . . . . . 14 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝐹‘𝑦) ∈ (Base‘𝑇) ∧ ((0g‘𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 1) → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥)))
126	96, 125	sylbid 242	. . . . . . . . . . . . 13 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥)))
127	126	adantld 493	. . . . . . . . . . . 12 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥)))
128	91, 127	sylbid 242	. . . . . . . . . . 11 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥)))
129	88, 128	imim12d 81	. . . . . . . . . 10 ⊢ ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥 → 𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))) → (((norm‘𝑆)‘𝑦) < 𝑥 → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥))))
130	129	ralimdva 3177	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (∀𝑦 ∈ (Base‘𝑆)(((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥 → 𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))) → ∀𝑦 ∈ (Base‘𝑆)(((norm‘𝑆)‘𝑦) < 𝑥 → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥))))
131		rpxr 12397	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ*)
132		blval 22995	. . . . . . . . . . . 12 ⊢ ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ (0g‘𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ ℝ*) → ((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) = {𝑦 ∈ (Base‘𝑆) ∣ ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥})
133	19, 52, 131, 132	syl2an3an 1418	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → ((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) = {𝑦 ∈ (Base‘𝑆) ∣ ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥})
134	133	sseq1d 3997	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ {𝑦 ∈ (Base‘𝑆) ∣ ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥} ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
135		rabss 4047	. . . . . . . . . 10 ⊢ ({𝑦 ∈ (Base‘𝑆) ∣ ((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥} ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ∀𝑦 ∈ (Base‘𝑆)(((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥 → 𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
136	134, 135	syl6bb 289	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ∀𝑦 ∈ (Base‘𝑆)(((0g‘𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥 → 𝑦 ∈ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))))
137		eqid 2821	. . . . . . . . . 10 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
138		nmhmcn.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑆)
139		nmhmcn.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
140	10	adantr 483	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑆 ∈ (NrmMod ∩ ℂMod))
141	21	adantr 483	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈ (NrmMod ∩ ℂMod))
142		rpreccl 12414	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
143	142	adantl 484	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
144	143	rpxrd 12431	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ*)
145		simpr 487	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
146		simpl3 1189	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → ℚ ⊆ 𝐵)
147	146	ad2antrr 724	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → ℚ ⊆ 𝐵)
148	137, 16, 78, 106, 138, 139, 140, 141, 103, 144, 145, 147	nmoleub2b 23721	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥) ↔ ∀𝑦 ∈ (Base‘𝑆)(((norm‘𝑆)‘𝑦) < 𝑥 → (((norm‘𝑇)‘(𝐹‘𝑦)) / 𝑥) ≤ (1 / 𝑥))))
149	130, 136, 148	3imtr4d 296	. . . . . . . 8 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → ((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥)))
150	73, 100, 54	3jca 1124	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
151	142	rpred 12430	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
152	137	bddnghm 23334	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ ((1 / 𝑥) ∈ ℝ ∧ ((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
153	152	expr 459	. . . . . . . . 9 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (1 / 𝑥) ∈ ℝ) → (((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
154	150, 151, 153	syl2an 597	. . . . . . . 8 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
155	149, 154	syld 47	. . . . . . 7 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
156	155	rexlimdva 3284	. . . . . 6 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∃𝑥 ∈ ℝ+ ((0g‘𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (◡𝐹 “ ((0g‘𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
157	69, 156	mpd 15	. . . . 5 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 NGHom 𝑇))
158	157	ex 415	. . . 4 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
159	9, 158	impbid2 228	. . 3 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ 𝐹 ∈ (𝐽 Cn 𝐾)))
160	159	pm5.32da 581	. 2 ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
161	6, 160	bitrd 281	1 ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142   ∩ cin 3934   ⊆ wss 3935   class class class wbr 5065   × cxp 5552  ^◡ccnv 5553   ↾ cres 5556   “ cima 5557   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  ℝcr 10535  1c1 10537  ℝ^*cxr 10673   < clt 10674   ≤ cle 10675   / cdiv 11296  ℚcq 12347  ℝ⁺crp 12388  Basecbs 16482  Scalarcsca 16567  distcds 16573  TopOpenctopn 16694  0_gc0g 16712  Grpcgrp 18102   GrpHom cghm 18354  LModclmod 19633   LMHom clmhm 19790  ∞Metcxmet 20529  ballcbl 20531  MetOpencmopn 20534   Cn ccn 21831  ∞MetSpcxms 22926  MetSpcms 22927  normcnm 23185  NrmGrpcngp 23186  NrmModcnlm 23189   normOp cnmo 23313   NGHom cnghm 23314   NMHom cnmhm 23315  ℂModcclm 23665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-addf 10615  ax-mulf 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ico 12743  df-fz 12892  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-mulr 16578  df-starv 16579  df-tset 16583  df-ple 16584  df-ds 16586  df-unif 16587  df-0g 16714  df-topgen 16716  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-grp 18105  df-minusg 18106  df-sbg 18107  df-subg 18275  df-ghm 18355  df-cmn 18907  df-mgp 19239  df-ring 19298  df-cring 19299  df-subrg 19532  df-lmod 19635  df-lmhm 19793  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-mopn 20540  df-cnfld 20545  df-top 21501  df-topon 21518  df-topsp 21540  df-bases 21553  df-cn 21834  df-cnp 21835  df-xms 22929  df-ms 22930  df-nm 23191  df-ngp 23192  df-nlm 23195  df-nmo 23316  df-nghm 23317  df-nmhm 23318  df-clm 23666

This theorem is referenced by: (None)