Theorem nmhmcn 24292

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 4130	. . . 4 ⊢ (𝑆 ∈ (NrmMod ∩ ℂMod) → 𝑆 ∈ NrmMod)
2		elinel1 4130	. . . 4 ⊢ (𝑇 ∈ (NrmMod ∩ ℂMod) → 𝑇 ∈ NrmMod)
3		isnmhm 23919	. . . . 5 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
4	3	baib 536	. . . 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 1131	. 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 23918	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
10		simpll1 1211	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
11	10	elin1d 4133	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmMod)
12		nlmngp 23850	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
13		ngpms 23765	. . . . . . . . 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 23616	. . . . . . . 8 ⊢ (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
16		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		eqid 2739	. . . . . . . . 9 ⊢ ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
18	16, 17	xmsxmet 23618	. . . . . . . 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 485	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
21		simpll2 1212	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
22	21	elin1d 4133	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmMod)
23		nlmngp 23850	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
24		ngpms 23765	. . . . . . . . . . . . 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 23616	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
27		eqid 2739	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
28		eqid 2739	. . . . . . . . . . . . 13 ⊢ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
29	27, 28	xmsxmet 23618	. . . . . . . . . . . 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 23851	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ LMod)
32		eqid 2739	. . . . . . . . . . . . 13 ⊢ (0g‘𝑇) = (0g‘𝑇)
33	27, 32	lmod0vcl 20161	. . . . . . . . . . . 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 12743	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ+
36		rpxr 12748	. . . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
39	38	blopn 23665	. . . . . . . . . . 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 1370	. . . . . . . . . 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 23614	. . . . . . . . . . 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 2843	. . . . . . . . 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 22425	. . . . . . . . 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 23614	. . . . . . . . 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 2842	. . . . . . 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 23851	. . . . . . . . 9 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod)
50		eqid 2739	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
51	16, 50	lmod0vcl 20161	. . . . . . . . 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 20302	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
54	53	ad2antlr 724	. . . . . . . . . 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 18849	. . . . . . . . . 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 23575	. . . . . . . . . 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 1370	. . . . . . . . 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 2840	. . . . . . . 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 20305	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
62	61	ad2antlr 724	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
63		ffn 6609	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
64		elpreima 6944	. . . . . . . . 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 710	. . . . . . 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 2739	. . . . . . . 8 ⊢ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
68	67	mopni2 23658	. . . . . . 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 1370	. . . . . 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 1190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
71	70	elin1d 4133	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmGrp)
74	73	ad2antrr 723	. . . . . . . . . . . . . . . 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 23764	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 16 ⊢ (norm‘𝑆) = (norm‘𝑆)
79		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (dist‘𝑆) = (dist‘𝑆)
80	78, 16, 50, 79, 17	nmval2 23757	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 23509	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 5085	. . . . . . . . . . . 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 228	. . . . . . . . . . 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 723	. . . . . . . . . . . . 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 6944	. . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 23550	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
98	97	elin1d 4133	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmGrp)
101	100	ad2antrr 723	. . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
103	102	adantr 481	. . . . . . . . . . . . . . . . . . . 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 6970	. . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (norm‘𝑇) = (norm‘𝑇)
107	27, 106	nmcl 23781	. . . . . . . . . . . . . . . . . 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 10984	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
110		ltle 11072	. . . . . . . . . . . . . . . . 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 23764	. . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dist‘𝑇) = (dist‘𝑇)
115	106, 27, 32, 114, 28	nmval2 23757	. . . . . . . . . . . . . . . . . . 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 23509	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . 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 5085	. . . . . . . . . . . . . . . 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 10985	. . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . 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 12827	. . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . 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 239	. . . . . . . . . . . . 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 491	. . . . . . . . . . . 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 239	. . . . . . . . . . 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 3109	. . . . . . . . 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 12748	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ*)
132		blval 23548	. . . . . . . . . . . 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 1421	. . . . . . . . . . 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 3953	. . . . . . . . . 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 4006	. . . . . . . . . 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 2739	. . . . . . . . . 10 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
138		nmhmcn.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑆)
139		nmhmcn.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
140	10	adantr 481	. . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈ (NrmMod ∩ ℂMod))
142		rpreccl 12765	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
143	142	adantl 482	. . . . . . . . . . 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 12782	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ*)
145		simpr 485	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
146		simpl3 1192	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → ℚ ⊆ 𝐵)
147	146	ad2antrr 723	. . . . . . . . . 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 24290	. . . . . . . . 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 1127	. . . . . . . . 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 12781	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
152	137	bddnghm 23899	. . . . . . . . . 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 457	. . . . . . . . 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 3214	. . . . . 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 413	. . . 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 225	. . 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 278	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3065  ∃wrex 3066  {crab 3069   ∩ cin 3887   ⊆ wss 3888   class class class wbr 5075   × cxp 5588  ^◡ccnv 5589   ↾ cres 5592   “ cima 5593   Fn wfn 6432  ⟶wf 6433  ‘cfv 6437  (class class class)co 7284  ℝcr 10879  1c1 10881  ℝ^*cxr 11017   < clt 11018   ≤ cle 11019   / cdiv 11641  ℚcq 12697  ℝ⁺crp 12739  Basecbs 16921  Scalarcsca 16974  distcds 16980  TopOpenctopn 17141  0_gc0g 17159  Grpcgrp 18586   GrpHom cghm 18840  LModclmod 20132   LMHom clmhm 20290  ∞Metcxmet 20591  ballcbl 20593  MetOpencmopn 20596   Cn ccn 22384  ∞MetSpcxms 23479  MetSpcms 23480  normcnm 23741  NrmGrpcngp 23742  NrmModcnlm 23745   normOp cnmo 23878   NGHom cnghm 23879   NMHom cnmhm 23880  ℂModcclm 24234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958  ax-addf 10959  ax-mulf 10960

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-map 8626  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-sup 9210  df-inf 9211  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-4 12047  df-5 12048  df-6 12049  df-7 12050  df-8 12051  df-9 12052  df-n0 12243  df-z 12329  df-dec 12447  df-uz 12592  df-q 12698  df-rp 12740  df-xneg 12857  df-xadd 12858  df-xmul 12859  df-ico 13094  df-fz 13249  df-seq 13731  df-exp 13792  df-cj 14819  df-re 14820  df-im 14821  df-sqrt 14955  df-abs 14956  df-struct 16857  df-sets 16874  df-slot 16892  df-ndx 16904  df-base 16922  df-ress 16951  df-plusg 16984  df-mulr 16985  df-starv 16986  df-tset 16990  df-ple 16991  df-ds 16993  df-unif 16994  df-0g 17161  df-topgen 17163  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-grp 18589  df-minusg 18590  df-sbg 18591  df-subg 18761  df-ghm 18841  df-cmn 19397  df-mgp 19730  df-ring 19794  df-cring 19795  df-subrg 20031  df-lmod 20134  df-lmhm 20293  df-psmet 20598  df-xmet 20599  df-met 20600  df-bl 20601  df-mopn 20602  df-cnfld 20607  df-top 22052  df-topon 22069  df-topsp 22091  df-bases 22105  df-cn 22387  df-cnp 22388  df-xms 23482  df-ms 23483  df-nm 23747  df-ngp 23748  df-nlm 23751  df-nmo 23881  df-nghm 23882  df-nmhm 23883  df-clm 24235

This theorem is referenced by: (None)