Theorem nmhmcn 25027

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 4167	. . . 4 ⊢ (𝑆 ∈ (NrmMod ∩ ℂMod) → 𝑆 ∈ NrmMod)
2		elinel1 4167	. . . 4 ⊢ (𝑇 ∈ (NrmMod ∩ ℂMod) → 𝑇 ∈ NrmMod)
3		isnmhm 24641	. . . . 5 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
4	3	baib 535	. . . 4 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
5	1, 2, 4	syl2an 596	. . 3 ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
6	5	3adant3 1132	. 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 24640	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
10		simpll1 1213	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
11	10	elin1d 4170	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmMod)
12		nlmngp 24572	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
13		ngpms 24495	. . . . . . . . 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 24349	. . . . . . . 8 ⊢ (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
16		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		eqid 2730	. . . . . . . . 9 ⊢ ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
18	16, 17	xmsxmet 24351	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
21		simpll2 1214	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
22	21	elin1d 4170	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmMod)
23		nlmngp 24572	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
24		ngpms 24495	. . . . . . . . . . . . 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 24349	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
27		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
28		eqid 2730	. . . . . . . . . . . . 13 ⊢ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
29	27, 28	xmsxmet 24351	. . . . . . . . . . . 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 24573	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ LMod)
32		eqid 2730	. . . . . . . . . . . . 13 ⊢ (0g‘𝑇) = (0g‘𝑇)
33	27, 32	lmod0vcl 20804	. . . . . . . . . . . 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 12962	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ+
36		rpxr 12968	. . . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
39	38	blopn 24395	. . . . . . . . . . 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 1373	. . . . . . . . . 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 24347	. . . . . . . . . . 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 2832	. . . . . . . . 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 23159	. . . . . . . . 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 584	. . . . . . . 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 24347	. . . . . . . . 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 2831	. . . . . . 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 24573	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod)
50		eqid 2730	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
51	16, 50	lmod0vcl 20804	. . . . . . . . 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 20945	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
54	53	ad2antlr 727	. . . . . . . . . 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 19161	. . . . . . . . . 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 24308	. . . . . . . . . 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 1373	. . . . . . . . 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 2829	. . . . . . . 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 20948	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
62	61	ad2antlr 727	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
63		ffn 6691	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
64		elpreima 7033	. . . . . . . . 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 713	. . . . . . 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 2730	. . . . . . . 8 ⊢ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
68	67	mopni2 24388	. . . . . . 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 1373	. . . . . 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 1192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
71	70	elin1d 4170	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmGrp)
74	73	ad2antrr 726	. . . . . . . . . . . . . . . 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 24494	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (norm‘𝑆) = (norm‘𝑆)
79		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (dist‘𝑆) = (dist‘𝑆)
80	78, 16, 50, 79, 17	nmval2 24487	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑦) = (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g‘𝑆)))
81	76, 77, 80	syl2anc 584	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 24242	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 5120	. . . . . . . . . . . 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 229	. . . . . . . . . . 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 726	. . . . . . . . . . . . 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 7033	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 24283	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
98	97	elin1d 4170	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmGrp)
101	100	ad2antrr 726	. . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (norm‘𝑇) = (norm‘𝑇)
107	27, 106	nmcl 24511	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑦)) ∈ ℝ)
108	101, 105, 107	syl2anc 584	. . . . . . . . . . . . . . . . 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 11181	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
110		ltle 11269	. . . . . . . . . . . . . . . . 17 ⊢ ((((norm‘𝑇)‘(𝐹‘𝑦)) ∈ ℝ ∧ 1 ∈ ℝ) → (((norm‘𝑇)‘(𝐹‘𝑦)) < 1 → ((norm‘𝑇)‘(𝐹‘𝑦)) ≤ 1))
111	108, 109, 110	sylancl 586	. . . . . . . . . . . . . . . 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 24494	. . . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dist‘𝑇) = (dist‘𝑇)
115	106, 27, 32, 114, 28	nmval2 24487	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 24242	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . 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 5120	. . . . . . . . . . . . . . . 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 11182	. . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . 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 13048	. . . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 240	. . . . . . . . . . . . 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 490	. . . . . . . . . . . 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 240	. . . . . . . . . . 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 3146	. . . . . . . . 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 12968	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ*)
132		blval 24281	. . . . . . . . . . . 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 1424	. . . . . . . . . . 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 3981	. . . . . . . . . 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 4038	. . . . . . . . . 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	bitrdi 287	. . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
138		nmhmcn.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑆)
139		nmhmcn.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
140	10	adantr 480	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈ (NrmMod ∩ ℂMod))
142		rpreccl 12986	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
143	142	adantl 481	. . . . . . . . . . 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 13003	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ*)
145		simpr 484	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
146		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → ℚ ⊆ 𝐵)
147	146	ad2antrr 726	. . . . . . . . . 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 25025	. . . . . . . . 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 294	. . . . . . . 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 1128	. . . . . . . . 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 13002	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
152	137	bddnghm 24621	. . . . . . . . . 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 456	. . . . . . . . 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 596	. . . . . . . 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 3135	. . . . . 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 412	. . . 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 226	. . 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 579	. 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 279	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  {crab 3408   ∩ cin 3916   ⊆ wss 3917   class class class wbr 5110   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℝcr 11074  1c1 11076  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   / cdiv 11842  ℚcq 12914  ℝ⁺crp 12958  Basecbs 17186  Scalarcsca 17230  distcds 17236  TopOpenctopn 17391  0_gc0g 17409  Grpcgrp 18872   GrpHom cghm 19151  LModclmod 20773   LMHom clmhm 20933  ∞Metcxmet 21256  ballcbl 21258  MetOpencmopn 21261   Cn ccn 23118  ∞MetSpcxms 24212  MetSpcms 24213  normcnm 24471  NrmGrpcngp 24472  NrmModcnlm 24475   normOp cnmo 24600   NGHom cnghm 24601   NMHom cnmhm 24602  ℂModcclm 24969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ico 13319  df-fz 13476  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-0g 17411  df-topgen 17413  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-ghm 19152  df-cmn 19719  df-mgp 20057  df-ring 20151  df-cring 20152  df-subrg 20486  df-lmod 20775  df-lmhm 20936  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cn 23121  df-cnp 23122  df-xms 24215  df-ms 24216  df-nm 24477  df-ngp 24478  df-nlm 24481  df-nmo 24603  df-nghm 24604  df-nmhm 24605  df-clm 24970

This theorem is referenced by: (None)