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Theorem nlmvscn 24853

Description: The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 24856 and nlmtlm 24860. (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 24844	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
2		eqid 2763	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		nlmvscn.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4		eqid 2763	. . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹)
5		nlmvscn.sf	. . . . 5 ⊢ · = ( ·_sf ‘𝑊)
6	2, 3, 4, 5	lmodscaf 21014	. . . 4 ⊢ (𝑊 ∈ LMod → · :((Base‘𝐹) × (Base‘𝑊))⟶(Base‘𝑊))
7	1, 6	syl 18	. . 3 ⊢ (𝑊 ∈ NrmMod → · :((Base‘𝐹) × (Base‘𝑊))⟶(Base‘𝑊))
8		eqid 2763	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
9		eqid 2763	. . . . . . 7 ⊢ (dist‘𝐹) = (dist‘𝐹)
10		eqid 2763	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
11		eqid 2763	. . . . . . 7 ⊢ (norm‘𝐹) = (norm‘𝐹)
12		eqid 2763	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
13		eqid 2763	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1))
14		eqid 2763	. . . . . . 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 778	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ NrmMod)
16		simpr 489	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
17		simplrl 788	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝐹))
18		simplrr 789	. . . . . . 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 24852	. . . . . 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 3157	. . . . 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 788	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐹))
22		simprl 782	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝐹))
23	21, 22	ovresd 7577	. . . . . . . . . . 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 5119	. . . . . . . . . 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 789	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
26		simprr 784	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
27	25, 26	ovresd 7577	. . . . . . . . . . 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 5119	. . . . . . . . . 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 643	. . . . . . . . 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 21011	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥 · 𝑦) = (𝑥( ·_𝑠 ‘𝑊)𝑦))
31	30	ad2antlr 739	. . . . . . . . . . . 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 21011	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧 · 𝑤) = (𝑧( ·_𝑠 ‘𝑊)𝑤))
33	32	adantl 486	. . . . . . . . . . . 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 7428	. . . . . . . . . . 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 738	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
36	2, 3, 12, 4	lmodvscl 21008	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·_𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
37	35, 21, 25, 36	syl3anc 1398	. . . . . . . . . . . 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 21008	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧( ·_𝑠 ‘𝑊)𝑤) ∈ (Base‘𝑊))
39	35, 22, 26, 38	syl3anc 1398	. . . . . . . . . . . 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 7577	. . . . . . . . . . 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 2798	. . . . . . . . . 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 5119	. . . . . . . . 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 347	. . . . . . . 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 3227	. . . . . . 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 3189	. . . . . 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 3188	. . . . 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 260	. . . 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 3208	. . 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 24846	. . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
50		ngpms 24766	. . . . . 6 ⊢ (𝐹 ∈ NrmGrp → 𝐹 ∈ MetSp)
51	49, 50	syl 18	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ MetSp)
52		msxms 24620	. . . . 5 ⊢ (𝐹 ∈ MetSp → 𝐹 ∈ ∞MetSp)
53		eqid 2763	. . . . . 6 ⊢ ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) = ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))
54	4, 53	xmsxmet 24622	. . . . 5 ⊢ (𝐹 ∈ ∞MetSp → ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) ∈ (∞Met‘(Base‘𝐹)))
55	51, 52, 54	3syl 19	. . . 4 ⊢ (𝑊 ∈ NrmMod → ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) ∈ (∞Met‘(Base‘𝐹)))
56		nlmngp 24843	. . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
57		ngpms 24766	. . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
58	56, 57	syl 18	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ MetSp)
59		msxms 24620	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
60		eqid 2763	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
61	2, 60	xmsxmet 24622	. . . . 5 ⊢ (𝑊 ∈ ∞MetSp → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
62	58, 59, 61	3syl 19	. . . 4 ⊢ (𝑊 ∈ NrmMod → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
63		eqid 2763	. . . . 5 ⊢ (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))))
64		eqid 2763	. . . . 5 ⊢ (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))
65	63, 64, 64	txmetcn 24714	. . . 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 1398	. . 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 725	. 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 24618	. . . . 5 ⊢ (𝐹 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))))
70	51, 69	syl 18	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝐾 = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))))
71		nlmvscn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
72	71, 2, 60	mstopn 24618	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
73	58, 72	syl 18	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
74	70, 73	oveq12d 7428	. . 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 7428	. 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 2866	1 ⊢ (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 class class class wbr 5109 × cxp 5659 ↾ cres 5663 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1c1 11105 + caddc 11107 < clt 11247 / cdiv 11875 2c2 12299 ℝ⁺crp 13020 Basecbs 17273 Scalarcsca 17317 ·_𝑠 cvsca 17318 distcds 17323 TopOpenctopn 17478 LModclmod 20990 ·_sf cscaf 20991 ∞Metcxmet 21516 MetOpencmopn 21521 Cn ccn 23390 ×_t ctx 23726 ∞MetSpcxms 24483 MetSpcms 24484 normcnm 24742 NrmGrpcngp 24743 NrmModcnlm 24746

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-icc 13383 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-xrs 17560 df-qtop 17565 df-imas 17566 df-xps 17568 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-mulg 19138 df-cntz 19391 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-lmod 20992 df-scaf 20993 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cn 23393 df-cnp 23394 df-tx 23728 df-hmeo 23921 df-xms 24486 df-ms 24487 df-tms 24488 df-nm 24748 df-ngp 24749 df-nrg 24751 df-nlm 24752

This theorem is used by: nrgtrg 24856 nlmtlm 24860

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