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Theorem nmhmcn 25167
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.
StepHypRef Expression
1 elinel1 4211 . . . 4 (𝑆 ∈ (NrmMod ∩ ℂMod) → 𝑆 ∈ NrmMod)
2 elinel1 4211 . . . 4 (𝑇 ∈ (NrmMod ∩ ℂMod) → 𝑇 ∈ NrmMod)
3 isnmhm 24783 . . . . 5 (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
43baib 535 . . . 4 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
51, 2, 4syl2an 596 . . 3 ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
653adant3 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)
97, 8nghmcn 24782 . . . 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))
1110elin1d 4214 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmMod)
12 nlmngp 24714 . . . . . . . . 9 (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
13 ngpms 24629 . . . . . . . . 9 (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
1411, 12, 133syl 18 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ MetSp)
15 msxms 24480 . . . . . . . 8 (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
16 eqid 2735 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
17 eqid 2735 . . . . . . . . 9 ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
1816, 17xmsxmet 24482 . . . . . . . 8 (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
1914, 15, 183syl 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 1212 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
2221elin1d 4214 . . . . . . . . . . . . 13 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmMod)
23 nlmngp 24714 . . . . . . . . . . . . 13 (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
24 ngpms 24629 . . . . . . . . . . . . 13 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
2522, 23, 243syl 18 . . . . . . . . . . . 12 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ MetSp)
26 msxms 24480 . . . . . . . . . . . 12 (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
27 eqid 2735 . . . . . . . . . . . . 13 (Base‘𝑇) = (Base‘𝑇)
28 eqid 2735 . . . . . . . . . . . . 13 ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
2927, 28xmsxmet 24482 . . . . . . . . . . . 12 (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
3025, 26, 293syl 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 24715 . . . . . . . . . . . 12 (𝑇 ∈ NrmMod → 𝑇 ∈ LMod)
32 eqid 2735 . . . . . . . . . . . . 13 (0g𝑇) = (0g𝑇)
3327, 32lmod0vcl 20906 . . . . . . . . . . . 12 (𝑇 ∈ LMod → (0g𝑇) ∈ (Base‘𝑇))
3422, 31, 333syl 18 . . . . . . . . . . 11 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g𝑇) ∈ (Base‘𝑇))
35 1rp 13036 . . . . . . . . . . . 12 1 ∈ ℝ+
36 rpxr 13042 . . . . . . . . . . . 12 (1 ∈ ℝ+ → 1 ∈ ℝ*)
3735, 36mp1i 13 . . . . . . . . . . 11 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 1 ∈ ℝ*)
38 eqid 2735 . . . . . . . . . . . 12 (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
3938blopn 24529 . . . . . . . . . . 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)))))
4030, 34, 37, 39syl3anc 1370 . . . . . . . . . 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)))))
418, 27, 28mstopn 24478 . . . . . . . . . . 11 (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
4222, 23, 24, 414syl 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)))))
4340, 42eleqtrrd 2842 . . . . . . . . 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)
44 cnima 23289 . . . . . . . . 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)
4520, 43, 44syl2anc 584 . . . . . . . 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)
467, 16, 17mstopn 24478 . . . . . . . . 9 (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
4711, 12, 13, 464syl 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)))))
4845, 47eleqtrd 2841 . . . . . . 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)))))
49 nlmlmod 24715 . . . . . . . . 9 (𝑆 ∈ NrmMod → 𝑆 ∈ LMod)
50 eqid 2735 . . . . . . . . . 10 (0g𝑆) = (0g𝑆)
5116, 50lmod0vcl 20906 . . . . . . . . 9 (𝑆 ∈ LMod → (0g𝑆) ∈ (Base‘𝑆))
5211, 49, 513syl 18 . . . . . . . 8 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (0g𝑆) ∈ (Base‘𝑆))
53 lmghm 21048 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
5453ad2antlr 727 . . . . . . . . . 10 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
5550, 32ghmid 19253 . . . . . . . . . 10 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
5654, 55syl 17 . . . . . . . . 9 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹‘(0g𝑆)) = (0g𝑇))
5735a1i 11 . . . . . . . . . 10 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 1 ∈ ℝ+)
58 blcntr 24439 . . . . . . . . . 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))
5930, 34, 57, 58syl3anc 1370 . . . . . . . . 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))
6056, 59eqeltrd 2839 . . . . . . . 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))
6116, 27lmhmf 21051 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6261ad2antlr 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 6737 . . . . . . . . 9 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
64 elpreima 7078 . . . . . . . . 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))))
6562, 63, 643syl 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))))
6652, 60, 65mpbir2and 713 . . . . . . 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)))
67 eqid 2735 . . . . . . . 8 (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
6867mopni2 24522 . . . . . . 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)))
6919, 48, 66, 68syl3anc 1370 . . . . . 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)))
70 simpl1 1190 . . . . . . . . . . . . . . . . . . . 20 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ (NrmMod ∩ ℂMod))
7170elin1d 4214 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ NrmMod)
7271, 12syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ NrmGrp)
7372adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑆 ∈ NrmGrp)
7473ad2antrr 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 24628 . . . . . . . . . . . . . . . 16 (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp)
7674, 75syl 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 2735 . . . . . . . . . . . . . . . 16 (norm‘𝑆) = (norm‘𝑆)
79 eqid 2735 . . . . . . . . . . . . . . . 16 (dist‘𝑆) = (dist‘𝑆)
8078, 16, 50, 79, 17nmval2 24621 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑦) = (𝑦((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))(0g𝑆)))
8176, 77, 80syl2anc 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)))(0g𝑆)))
8219ad2antrr 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)))
8352ad2antrr 726 . . . . . . . . . . . . . . 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))
84 xmetsym 24373 . . . . . . . . . . . . . . 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))
8582, 77, 83, 84syl3anc 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)))(0g𝑆)) = ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦))
8681, 85eqtrd 2775 . . . . . . . . . . . . 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))
8786breq1d 5158 . . . . . . . . . . . 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))
8887biimpd 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 → ((0g𝑆)((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑥))
8962ad2antrr 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 7078 . . . . . . . . . . . . 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))))
9189, 63, 903syl 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))))
9230ad2antrr 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)))
9334ad2antrr 726 . . . . . . . . . . . . . . 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))
9435, 36mp1i 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 24414 . . . . . . . . . . . . . . 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)))
9692, 93, 94, 95syl3anc 1370 . . . . . . . . . . . . . 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)))
97 simpl2 1191 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ (NrmMod ∩ ℂMod))
9897elin1d 4214 . . . . . . . . . . . . . . . . . . . . 21 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ NrmMod)
9998, 23syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ NrmGrp)
10099adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑇 ∈ NrmGrp)
101100ad2antrr 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 769 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
103102adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝐹 ∈ (𝑆 LMHom 𝑇))
104103, 61syl 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))
105104ffvelcdmda 7104 . . . . . . . . . . . . . . . . . 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))
106 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (norm‘𝑇) = (norm‘𝑇)
10727, 106nmcl 24645 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑦)) ∈ ℝ)
108101, 105, 107syl2anc 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)‘(𝐹𝑦)) ∈ ℝ)
109 1re 11259 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
110 ltle 11347 . . . . . . . . . . . . . . . . 17 ((((norm‘𝑇)‘(𝐹𝑦)) ∈ ℝ ∧ 1 ∈ ℝ) → (((norm‘𝑇)‘(𝐹𝑦)) < 1 → ((norm‘𝑇)‘(𝐹𝑦)) ≤ 1))
111108, 109, 110sylancl 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)‘(𝐹𝑦)) < 1 → ((norm‘𝑇)‘(𝐹𝑦)) ≤ 1))
112 ngpgrp 24628 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
113101, 112syl 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 2735 . . . . . . . . . . . . . . . . . . . 20 (dist‘𝑇) = (dist‘𝑇)
115106, 27, 32, 114, 28nmval2 24621 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Grp ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑦)) = ((𝐹𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g𝑇)))
116113, 105, 115syl2anc 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)‘(𝐹𝑦)) = ((𝐹𝑦)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(0g𝑇)))
117 xmetsym 24373 . . . . . . . . . . . . . . . . . . 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)))
11892, 105, 93, 117syl3anc 1370 . . . . . . . . . . . . . . . . . 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)))
119116, 118eqtrd 2775 . . . . . . . . . . . . . . . . 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)))
120119breq1d 5158 . . . . . . . . . . . . . . . 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))
121 1red 11260 . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 17 ((((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ ℝ+)
123108, 121, 122lediv1d 13121 . . . . . . . . . . . . . . . 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)))
124111, 120, 1233imtr3d 293 . . . . . . . . . . . . . . 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)))
125124adantld 490 . . . . . . . . . . . . . 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)))
12696, 125sylbid 240 . . . . . . . . . . . . 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)))
127126adantld 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) ∧ (𝐹𝑦) ∈ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
12891, 127sylbid 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 “ ((0g𝑇)(ball‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))1)) → (((norm‘𝑇)‘(𝐹𝑦)) / 𝑥) ≤ (1 / 𝑥)))
12988, 128imim12d 81 . . . . . . . . . 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))))
130129ralimdva 3165 . . . . . . . . 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))))
131 rpxr 13042 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
132 blval 24412 . . . . . . . . . . . 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})
13319, 52, 131, 132syl2an3an 1421 . . . . . . . . . . 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})
134133sseq1d 4027 . . . . . . . . . 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))))
135 rabss 4082 . . . . . . . . . 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))))
136134, 135bitrdi 287 . . . . . . . . 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)))))
137 eqid 2735 . . . . . . . . . 10 (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
138 nmhmcn.g . . . . . . . . . 10 𝐺 = (Scalar‘𝑆)
139 nmhmcn.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
14010adantr 480 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → 𝑆 ∈ (NrmMod ∩ ℂMod))
14121adantr 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 13059 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
143142adantl 481 . . . . . . . . . . 11 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
144143rpxrd 13076 . . . . . . . . . 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 1192 . . . . . . . . . . 11 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → ℚ ⊆ 𝐵)
147146ad2antrr 726 . . . . . . . . . 10 (((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ℝ+) → ℚ ⊆ 𝐵)
148137, 16, 78, 106, 138, 139, 140, 141, 103, 144, 145, 147nmoleub2b 25165 . . . . . . . . 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))))
149130, 136, 1483imtr4d 294 . . . . . . . 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)))
15073, 100, 543jca 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)))
151142rpred 13075 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
152137bddnghm 24763 . . . . . . . . . 10 (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ ((1 / 𝑥) ∈ ℝ ∧ ((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
153152expr 456 . . . . . . . . 9 (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (1 / 𝑥) ∈ ℝ) → (((𝑆 normOp 𝑇)‘𝐹) ≤ (1 / 𝑥) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
154150, 151, 153syl2an 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)))
155149, 154syld 47 . . . . . . 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)))
156155rexlimdva 3153 . . . . . 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)))
15769, 156mpd 15 . . . . 5 ((((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝑆 NGHom 𝑇))
158157ex 412 . . . 4 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ (𝑆 NGHom 𝑇)))
1599, 158impbid2 226 . . 3 (((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ 𝐹 ∈ (𝐽 Cn 𝐾)))
160159pm5.32da 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))))
1616, 160bitrd 279 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cin 3962  wss 3963   class class class wbr 5148   × cxp 5687  ccnv 5688  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cr 11152  1c1 11154  *cxr 11292   < clt 11293  cle 11294   / cdiv 11918  cq 12988  +crp 13032  Basecbs 17245  Scalarcsca 17301  distcds 17307  TopOpenctopn 17468  0gc0g 17486  Grpcgrp 18964   GrpHom cghm 19243  LModclmod 20875   LMHom clmhm 21036  ∞Metcxmet 21367  ballcbl 21369  MetOpencmopn 21372   Cn ccn 23248  ∞MetSpcxms 24343  MetSpcms 24344  normcnm 24605  NrmGrpcngp 24606  NrmModcnlm 24609   normOp cnmo 24742   NGHom cnghm 24743   NMHom cnmhm 24744  ℂModcclm 25109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ico 13390  df-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-0g 17488  df-topgen 17490  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-cmn 19815  df-mgp 20153  df-ring 20253  df-cring 20254  df-subrg 20587  df-lmod 20877  df-lmhm 21039  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-xms 24346  df-ms 24347  df-nm 24611  df-ngp 24612  df-nlm 24615  df-nmo 24745  df-nghm 24746  df-nmhm 24747  df-clm 25110
This theorem is referenced by: (None)
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