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Theorem nlmvscn 24749

Description: The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 24752 and nlmtlm 24756. (Contributed by Mario Carneiro, 4-Oct-2015.)

**Hypotheses**
Ref	Expression
nlmvscn.f	⊢ 𝐹 = (Scalar‘𝑊)
nlmvscn.sf	⊢ · = ( ·_sf ‘𝑊)
nlmvscn.j	⊢ 𝐽 = (TopOpen‘𝑊)
nlmvscn.kf	⊢ 𝐾 = (TopOpen‘𝐹)

**Assertion**
Ref	Expression
nlmvscn	⊢ (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))

Proof of Theorem nlmvscn Dummy variables 𝑟 𝑥 𝑦 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nlmlmod 24740	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
2		eqid 2764	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		nlmvscn.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4		eqid 2764	. . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹)
5		nlmvscn.sf	. . . . 5 ⊢ · = ( ·_sf ‘𝑊)
6	2, 3, 4, 5	lmodscaf 20953	. . . 4 ⊢ (𝑊 ∈ LMod → · :((Base‘𝐹) × (Base‘𝑊))⟶(Base‘𝑊))
7	1, 6	syl 17	. . 3 ⊢ (𝑊 ∈ NrmMod → · :((Base‘𝐹) × (Base‘𝑊))⟶(Base‘𝑊))
8		eqid 2764	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
9		eqid 2764	. . . . . . 7 ⊢ (dist‘𝐹) = (dist‘𝐹)
10		eqid 2764	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
11		eqid 2764	. . . . . . 7 ⊢ (norm‘𝐹) = (norm‘𝐹)
12		eqid 2764	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
13		eqid 2764	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1))
14		eqid 2764	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑦) + ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1)))) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑦) + ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1))))
15		simpll 776	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ NrmMod)
16		simpr 488	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
17		simplrl 786	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝐹))
18		simplrr 787	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (Base‘𝑊))
19	3, 2, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	nlmvscnlem1 24748	. . . . . 6 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟))
20	19	ralrimiva 3156	. . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) → ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟))
21		simplrl 786	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐹))
22		simprl 780	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝐹))
23	21, 22	ovresd 7565	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) = (𝑥(dist‘𝐹)𝑧))
24	23	breq1d 5112	. . . . . . . . . 10 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ↔ (𝑥(dist‘𝐹)𝑧) < 𝑠))
25		simplrr 787	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
26		simprr 782	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
27	25, 26	ovresd 7565	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) = (𝑦(dist‘𝑊)𝑤))
28	27	breq1d 5112	. . . . . . . . . 10 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠 ↔ (𝑦(dist‘𝑊)𝑤) < 𝑠))
29	24, 28	anbi12d 641	. . . . . . . . 9 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) ↔ ((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠)))
30	2, 3, 4, 5, 12	scafval 20950	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥 · 𝑦) = (𝑥( ·_𝑠 ‘𝑊)𝑦))
31	30	ad2antlr 737	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 · 𝑦) = (𝑥( ·_𝑠 ‘𝑊)𝑦))
32	2, 3, 4, 5, 12	scafval 20950	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧 · 𝑤) = (𝑧( ·_𝑠 ‘𝑊)𝑤))
33	32	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 · 𝑤) = (𝑧( ·_𝑠 ‘𝑊)𝑤))
34	31, 33	oveq12d 7416	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) = ((𝑥( ·_𝑠 ‘𝑊)𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧( ·_𝑠 ‘𝑊)𝑤)))
35	1	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
36	2, 3, 12, 4	lmodvscl 20947	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·_𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
37	35, 21, 25, 36	syl3anc 1392	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥( ·_𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
38	2, 3, 12, 4	lmodvscl 20947	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧( ·_𝑠 ‘𝑊)𝑤) ∈ (Base‘𝑊))
39	35, 22, 26, 38	syl3anc 1392	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧( ·_𝑠 ‘𝑊)𝑤) ∈ (Base‘𝑊))
40	37, 39	ovresd 7565	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧( ·_𝑠 ‘𝑊)𝑤)) = ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)))
41	34, 40	eqtrd 2799	. . . . . . . . . 10 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) = ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)))
42	41	breq1d 5112	. . . . . . . . 9 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → (((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟 ↔ ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟))
43	29, 42	imbi12d 346	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → ((((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟) ↔ (((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟)))
44	43	2ralbidva 3226	. . . . . . 7 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) → (∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟) ↔ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟)))
45	44	rexbidv 3188	. . . . . 6 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) → (∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟) ↔ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟)))
46	45	ralbidv 3187	. . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) → (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟) ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝐹)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → ((𝑥( ·_𝑠 ‘𝑊)𝑦)(dist‘𝑊)(𝑧( ·_𝑠 ‘𝑊)𝑤)) < 𝑟)))
47	20, 46	mpbird 259	. . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) → ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟))
48	47	ralrimivva 3207	. . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟))
49	3	nlmngp2 24742	. . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
50		ngpms 24662	. . . . . 6 ⊢ (𝐹 ∈ NrmGrp → 𝐹 ∈ MetSp)
51	49, 50	syl 17	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ MetSp)
52		msxms 24516	. . . . 5 ⊢ (𝐹 ∈ MetSp → 𝐹 ∈ ∞MetSp)
53		eqid 2764	. . . . . 6 ⊢ ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) = ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))
54	4, 53	xmsxmet 24518	. . . . 5 ⊢ (𝐹 ∈ ∞MetSp → ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) ∈ (∞Met‘(Base‘𝐹)))
55	51, 52, 54	3syl 18	. . . 4 ⊢ (𝑊 ∈ NrmMod → ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) ∈ (∞Met‘(Base‘𝐹)))
56		nlmngp 24739	. . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
57		ngpms 24662	. . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
58	56, 57	syl 17	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ MetSp)
59		msxms 24516	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
60		eqid 2764	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
61	2, 60	xmsxmet 24518	. . . . 5 ⊢ (𝑊 ∈ ∞MetSp → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
62	58, 59, 61	3syl 18	. . . 4 ⊢ (𝑊 ∈ NrmMod → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
63		eqid 2764	. . . . 5 ⊢ (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))))
64		eqid 2764	. . . . 5 ⊢ (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))
65	63, 64, 64	txmetcn 24610	. . . 4 ⊢ ((((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) ∈ (∞Met‘(Base‘𝐹)) ∧ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)) ∧ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊))) → ( · ∈ (((MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) ↔ ( · :((Base‘𝐹) × (Base‘𝑊))⟶(Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟))))
66	55, 62, 62, 65	syl3anc 1392	. . 3 ⊢ (𝑊 ∈ NrmMod → ( · ∈ (((MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) ↔ ( · :((Base‘𝐹) × (Base‘𝑊))⟶(Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝐹)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 · 𝑦)((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))(𝑧 · 𝑤)) < 𝑟))))
67	7, 48, 66	mpbir2and 723	. 2 ⊢ (𝑊 ∈ NrmMod → · ∈ (((MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
68		nlmvscn.kf	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝐹)
69	68, 4, 53	mstopn 24514	. . . . 5 ⊢ (𝐹 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))))
70	51, 69	syl 17	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝐾 = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))))
71		nlmvscn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
72	71, 2, 60	mstopn 24514	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
73	58, 72	syl 17	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
74	70, 73	oveq12d 7416	. . 3 ⊢ (𝑊 ∈ NrmMod → (𝐾 ×_t 𝐽) = ((MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
75	74, 73	oveq12d 7416	. 2 ⊢ (𝑊 ∈ NrmMod → ((𝐾 ×_t 𝐽) Cn 𝐽) = (((MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
76	67, 75	eleqtrrd 2867	1 ⊢ (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ∀wral 3078 ∃wrex 3088 class class class wbr 5102 × cxp 5647 ↾ cres 5651 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 1c1 11076 + caddc 11078 < clt 11218 / cdiv 11846 2c2 12274 ℝ⁺crp 12995 Basecbs 17247 Scalarcsca 17291 ·_𝑠 cvsca 17292 distcds 17297 TopOpenctopn 17452 LModclmod 20929 ·_sf cscaf 20930 ∞Metcxmet 21411 MetOpencmopn 21416 Cn ccn 23286 ×_t ctx 23622 ∞MetSpcxms 24379 MetSpcms 24380 normcnm 24638 NrmGrpcngp 24639 NrmModcnlm 24642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-grp 18980 df-minusg 18981 df-sbg 18982 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-abl 19825 df-mgp 20189 df-rng 20201 df-ur 20234 df-ring 20287 df-lmod 20931 df-scaf 20932 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cn 23289 df-cnp 23290 df-tx 23624 df-hmeo 23817 df-xms 24382 df-ms 24383 df-tms 24384 df-nm 24644 df-ngp 24645 df-nrg 24647 df-nlm 24648

This theorem is referenced by: nrgtrg 24752 nlmtlm 24756

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