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Theorem nmhmcn 22659
Description: A linear operator over a normed complex 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.
StepHypRef Expression
1 inss1 3794 . . . . 5 (NrmMod ∩ ℂMod) ⊆ NrmMod
21sseli 3563 . . . 4 (𝑆 ∈ (NrmMod ∩ ℂMod) → 𝑆 ∈ NrmMod)
31sseli 3563 . . . 4 (𝑇 ∈ (NrmMod ∩ ℂMod) → 𝑇 ∈ NrmMod)
4 isnmhm 22292 . . . . 5 (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
54baib 941 . . . 4 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
62, 3, 5syl2an 492 . . 3 ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
763adant3 1073 . 2 ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
8 nmhmcn.j . . . . 5 𝐽 = (TopOpen‘𝑆)
9 nmhmcn.k . . . . 5 𝐾 = (TopOpen‘𝑇)
108, 9nghmcn 22291 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
11 simpll1 1092 . . . . . . . . . 10 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
121, 11sseldi 3565 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmMod)
13 nlmngp 22224 . . . . . . . . 9 (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
14 ngpms 22155 . . . . . . . . 9 (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
1512, 13, 143syl 18 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ MetSp)
16 msxms 22010 . . . . . . . 8 (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
17 eqid 2609 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
18 eqid 2609 . . . . . . . . 9 ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
1917, 18xmsxmet 22012 . . . . . . . 8 (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
2015, 16, 193syl 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)))
21 simpr 475 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
22 simpll2 1093 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
231, 22sseldi 3565 . . . . . . . . . . . . 13 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmMod)
24 nlmngp 22224 . . . . . . . . . . . . 13 (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
25 ngpms 22155 . . . . . . . . . . . . 13 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
2623, 24, 253syl 18 . . . . . . . . . . . 12 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ MetSp)
27 msxms 22010 . . . . . . . . . . . 12 (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
28 eqid 2609 . . . . . . . . . . . . 13 (Base‘𝑇) = (Base‘𝑇)
29 eqid 2609 . . . . . . . . . . . . 13 ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
3028, 29xmsxmet 22012 . . . . . . . . . . . 12 (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
3126, 27, 303syl 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)))
32 nlmlmod 22225 . . . . . . . . . . . 12 (𝑇 ∈ NrmMod → 𝑇 ∈ LMod)
33 eqid 2609 . . . . . . . . . . . . 13 (0g𝑇) = (0g𝑇)
3428, 33lmod0vcl 18661 . . . . . . . . . . . 12 (𝑇 ∈ LMod → (0g𝑇) ∈ (Base‘𝑇))
3523, 32, 343syl 18 . . . . . . . . . . 11 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g𝑇) ∈ (Base‘𝑇))
36 1rp 11668 . . . . . . . . . . . 12 1 ∈ ℝ+
37 rpxr 11672 . . . . . . . . . . . 12 (1 ∈ ℝ+ → 1 ∈ ℝ*)
3836, 37mp1i 13 . . . . . . . . . . 11 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 1 ∈ ℝ*)
39 eqid 2609 . . . . . . . . . . . 12 (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
4039blopn 22056 . . . . . . . . . . 11 ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (0g𝑇) ∈ (Base‘𝑇) ∧ 1 ∈ ℝ*) → ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
4131, 35, 38, 40syl3anc 1317 . . . . . . . . . 10 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
429, 28, 29mstopn 22008 . . . . . . . . . . 11 (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
4323, 24, 25, 424syl 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)))))
4441, 43eleqtrrd 2690 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ 𝐾)
45 cnima 20821 . . . . . . . . 9 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ∈ 𝐾) → (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ 𝐽)
4621, 44, 45syl2anc 690 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ 𝐽)
478, 17, 18mstopn 22008 . . . . . . . . 9 (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
4812, 13, 14, 474syl 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)))))
4946, 48eleqtrd 2689 . . . . . . 7 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
50 nlmlmod 22225 . . . . . . . . 9 (𝑆 ∈ NrmMod → 𝑆 ∈ LMod)
51 eqid 2609 . . . . . . . . . 10 (0g𝑆) = (0g𝑆)
5217, 51lmod0vcl 18661 . . . . . . . . 9 (𝑆 ∈ LMod → (0g𝑆) ∈ (Base‘𝑆))
5312, 50, 523syl 18 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g𝑆) ∈ (Base‘𝑆))
54 lmghm 18798 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
5554ad2antlr 758 . . . . . . . . . 10 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
5651, 33ghmid 17435 . . . . . . . . . 10 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
5755, 56syl 17 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹‘(0g𝑆)) = (0g𝑇))
5836a1i 11 . . . . . . . . . 10 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 1 ∈ ℝ+)
59 blcntr 21969 . . . . . . . . . 10 ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (0g𝑇) ∈ (Base‘𝑇) ∧ 1 ∈ ℝ+) → (0g𝑇) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))
6031, 35, 58, 59syl3anc 1317 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g𝑇) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))
6157, 60eqeltrd 2687 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹‘(0g𝑆)) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))
6217, 28lmhmf 18801 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6362ad2antlr 758 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
64 ffn 5944 . . . . . . . . 9 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
65 elpreima 6230 . . . . . . . . 9 (𝐹 Fn (Base‘𝑆) → ((0g𝑆) ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
6663, 64, 653syl 18 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((0g𝑆) ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
6753, 61, 66mpbir2and 958 . . . . . . 7 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g𝑆) ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))
68 eqid 2609 . . . . . . . 8 (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
6968mopni2 22049 . . . . . . 7 ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ∈ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) ∧ (0g𝑆) ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))) → ∃𝑥 ∈ ℝ+ ((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))
7020, 49, 67, 69syl3anc 1317 . . . . . 6 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∃𝑥 ∈ ℝ+ ((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))
71 simpl1 1056 . . . . . . . . . . . . . . . . . . . 20 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
721, 71sseldi 3565 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ NrmMod)
7372, 13syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ NrmGrp)
7473adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmGrp)
7574ad2antrr 757 . . . . . . . . . . . . . . . 16 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ NrmGrp)
76 ngpgrp 22154 . . . . . . . . . . . . . . . 16 (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp)
7775, 76syl 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)
78 simpr 475 . . . . . . . . . . . . . . 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))
79 eqid 2609 . . . . . . . . . . . . . . . 16 (norm‘𝑆) = (norm‘𝑆)
80 eqid 2609 . . . . . . . . . . . . . . . 16 (dist‘𝑆) = (dist‘𝑆)
8179, 17, 51, 80, 18nmval2 22147 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑦) = (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g𝑆)))
8277, 78, 81syl2anc 690 . . . . . . . . . . . . . 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)))(0g𝑆)))
8320ad2antrr 757 . . . . . . . . . . . . . . 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)))
8453ad2antrr 757 . . . . . . . . . . . . . . 15 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (0g𝑆) ∈ (Base‘𝑆))
85 xmetsym 21903 . . . . . . . . . . . . . . 15 ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆) ∧ (0g𝑆) ∈ (Base‘𝑆)) → (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g𝑆)) = ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
8683, 78, 84, 85syl3anc 1317 . . . . . . . . . . . . . 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)))(0g𝑆)) = ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
8782, 86eqtrd 2643 . . . . . . . . . . . . 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) = ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
8887breq1d 4587 . . . . . . . . . . . 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 ↔ ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥))
8988biimpd 217 . . . . . . . . . . 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 → ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥))
9063ad2antrr 757 . . . . . . . . . . . . 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))
91 elpreima 6230 . . . . . . . . . . . . 13 (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
9290, 64, 913syl 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 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
9331ad2antrr 757 . . . . . . . . . . . . . . 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)))
9435ad2antrr 757 . . . . . . . . . . . . . . 15 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (0g𝑇) ∈ (Base‘𝑇))
9536, 37mp1i 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*)
96 elbl 21944 . . . . . . . . . . . . . . 15 ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (0g𝑇) ∈ (Base‘𝑇) ∧ 1 ∈ ℝ*) → ((𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ↔ ((𝐹𝑦) ∈ (Base‘𝑇) ∧ ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 1)))
9793, 94, 95, 96syl3anc 1317 . . . . . . . . . . . . . 14 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) ↔ ((𝐹𝑦) ∈ (Base‘𝑇) ∧ ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 1)))
98 simpl2 1057 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
991, 98sseldi 3565 . . . . . . . . . . . . . . . . . . . . 21 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ NrmMod)
10099, 24syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ NrmGrp)
101100adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmGrp)
102101ad2antrr 757 . . . . . . . . . . . . . . . . . 18 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ NrmGrp)
103 simplr 787 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
104103adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝐹 ∈ (𝑆 LMHom 𝑇))
105104, 62syl 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))
106105ffvelrnda 6252 . . . . . . . . . . . . . . . . . 18 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹𝑦) ∈ (Base‘𝑇))
107 eqid 2609 . . . . . . . . . . . . . . . . . . 19 (norm‘𝑇) = (norm‘𝑇)
10828, 107nmcl 22170 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑦)) ∈ ℝ)
109102, 106, 108syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹𝑦)) ∈ ℝ)
110 1re 9895 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
111 ltle 9977 . . . . . . . . . . . . . . . . 17 ((((norm‘𝑇)‘(𝐹𝑦)) ∈ ℝ ∧ 1 ∈ ℝ) → (((norm‘𝑇)‘(𝐹𝑦)) < 1 → ((norm‘𝑇)‘(𝐹𝑦)) ≤ 1))
112109, 110, 111sylancl 692 . . . . . . . . . . . . . . . 16 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑇)‘(𝐹𝑦)) < 1 → ((norm‘𝑇)‘(𝐹𝑦)) ≤ 1))
113 ngpgrp 22154 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
114102, 113syl 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)
115 eqid 2609 . . . . . . . . . . . . . . . . . . . 20 (dist‘𝑇) = (dist‘𝑇)
116107, 28, 33, 115, 29nmval2 22147 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Grp ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑦)) = ((𝐹𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g𝑇)))
117114, 106, 116syl2anc 690 . . . . . . . . . . . . . . . . . 18 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹𝑦)) = ((𝐹𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g𝑇)))
118 xmetsym 21903 . . . . . . . . . . . . . . . . . . 19 ((((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)) ∧ (𝐹𝑦) ∈ (Base‘𝑇) ∧ (0g𝑇) ∈ (Base‘𝑇)) → ((𝐹𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g𝑇)) = ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)))
11993, 106, 94, 118syl3anc 1317 . . . . . . . . . . . . . . . . . 18 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g𝑇)) = ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)))
120117, 119eqtrd 2643 . . . . . . . . . . . . . . . . 17 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹𝑦)) = ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)))
121120breq1d 4587 . . . . . . . . . . . . . . . 16 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑇)‘(𝐹𝑦)) < 1 ↔ ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 1))
122 1red 9911 . . . . . . . . . . . . . . . . 17 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 1 ∈ ℝ)
123 simplr 787 . . . . . . . . . . . . . . . . 17 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ ℝ+)
124109, 122, 123lediv1d 11750 . . . . . . . . . . . . . . . 16 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((norm‘𝑇)‘(𝐹𝑦)) ≤ 1 ↔ (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
125112, 121, 1243imtr3d 280 . . . . . . . . . . . . . . 15 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 1 → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
126125adantld 481 . . . . . . . . . . . . . 14 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝐹𝑦) ∈ (Base‘𝑇) ∧ ((0g𝑇)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 1) → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
12797, 126sylbid 228 . . . . . . . . . . . . 13 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1) → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
128127adantld 481 . . . . . . . . . . . 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) ∧ (𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
12992, 128sylbid 228 . . . . . . . . . . 11 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑦 ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
13089, 129imim12d 78 . . . . . . . . . 10 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → ((((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥𝑦 ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))) → (((norm‘𝑆)‘𝑦) < 𝑥 → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥))))
131130ralimdva 2944 . . . . . . . . 9 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (∀𝑦 ∈ (Base‘𝑆)(((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥𝑦 ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))) → ∀𝑦 ∈ (Base‘𝑆)(((norm‘𝑆)‘𝑦) < 𝑥 → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥))))
13220adantr 479 . . . . . . . . . . . 12 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
13353adantr 479 . . . . . . . . . . . 12 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (0g𝑆) ∈ (Base‘𝑆))
134 rpxr 11672 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
135134adantl 480 . . . . . . . . . . . 12 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ*)
136 blval 21942 . . . . . . . . . . . 12 ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ (0g𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ ℝ*) → ((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) = {𝑦 ∈ (Base‘𝑆) ∣ ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥})
137132, 133, 135, 136syl3anc 1317 . . . . . . . . . . 11 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → ((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) = {𝑦 ∈ (Base‘𝑆) ∣ ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥})
138137sseq1d 3594 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ {𝑦 ∈ (Base‘𝑆) ∣ ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥} ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
139 rabss 3641 . . . . . . . . . 10 ({𝑦 ∈ (Base‘𝑆) ∣ ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥} ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ∀𝑦 ∈ (Base‘𝑆)(((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥𝑦 ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1))))
140138, 139syl6bb 274 . . . . . . . . 9 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) ↔ ∀𝑦 ∈ (Base‘𝑆)(((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥𝑦 ∈ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)))))
141 eqid 2609 . . . . . . . . . 10 (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
142 nmhmcn.g . . . . . . . . . 10 𝐺 = (Scalar‘𝑆)
143 nmhmcn.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
14411adantr 479 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑆 ∈ (NrmMod ∩ ℂMod))
14522adantr 479 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈ (NrmMod ∩ ℂMod))
146 rpreccl 11689 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
147146adantl 480 . . . . . . . . . . 11 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
148147rpxrd 11705 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ*)
149 simpr 475 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
150 simpl3 1058 . . . . . . . . . . 11 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → ℚ ⊆ 𝐵)
151150ad2antrr 757 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → ℚ ⊆ 𝐵)
152141, 17, 79, 107, 142, 143, 144, 145, 104, 148, 149, 151nmoleub2b 22657 . . . . . . . . 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)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥))))
153131, 140, 1523imtr4d 281 . . . . . . . 8 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → ((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥)))
15474, 101, 553jca 1234 . . . . . . . . 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)))
155146rpred 11704 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
156141bddnghm 22272 . . . . . . . . . 10 (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ ((1 / 𝑥) ∈ ℝ ∧ ((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
157156expr 640 . . . . . . . . 9 (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (1 / 𝑥) ∈ ℝ) → (((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
158154, 155, 157syl2an 492 . . . . . . . 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)))
159153, 158syld 45 . . . . . . 7 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
160159rexlimdva 3012 . . . . . 6 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∃𝑥 ∈ ℝ+ ((0g𝑆)(ball‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))𝑥) ⊆ (𝐹 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
16170, 160mpd 15 . . . . 5 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 NGHom 𝑇))
162161ex 448 . . . 4 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
16310, 162impbid2 214 . . 3 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ 𝐹 ∈ (𝐽 Cn 𝐾)))
164163pm5.32da 670 . 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))))
1657, 164bitrd 266 1 ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  {crab 2899  cin 3538  wss 3539   class class class wbr 4577   × cxp 5026  ccnv 5027  cres 5030  cima 5031   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  cr 9791  1c1 9793  *cxr 9929   < clt 9930  cle 9931   / cdiv 10533  cq 11620  +crp 11664  Basecbs 15641  Scalarcsca 15717  distcds 15723  TopOpenctopn 15851  0gc0g 15869  Grpcgrp 17191   GrpHom cghm 17426  LModclmod 18632   LMHom clmhm 18786  ∞Metcxmt 19498  ballcbl 19500  MetOpencmopn 19503   Cn ccn 20780  ∞MetSpcxme 21873  MetSpcmt 21874  normcnm 22132  NrmGrpcngp 22133  NrmModcnlm 22136   normOp cnmo 22251   NGHom cnghm 22252   NMHom cnmhm 22253  ℂModcclm 22601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ico 12008  df-fz 12153  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-starv 15729  df-tset 15733  df-ple 15734  df-ds 15737  df-unif 15738  df-0g 15871  df-topgen 15873  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-minusg 17195  df-sbg 17196  df-subg 17360  df-ghm 17427  df-cmn 17964  df-mgp 18259  df-ring 18318  df-cring 18319  df-subrg 18547  df-lmod 18634  df-lmhm 18789  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-cnfld 19514  df-top 20463  df-bases 20464  df-topon 20465  df-topsp 20466  df-cn 20783  df-cnp 20784  df-xms 21876  df-ms 21877  df-nm 22138  df-ngp 22139  df-nlm 22142  df-nmo 22254  df-nghm 22255  df-nmhm 22256  df-clm 22602
This theorem is referenced by: (None)
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