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Theorem nlmvscn 24582

Description: The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 24585 and nlmtlm 24589. (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 24573	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
2		eqid 2730	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		nlmvscn.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4		eqid 2730	. . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹)
5		nlmvscn.sf	. . . . 5 ⊢ · = ( ·_sf ‘𝑊)
6	2, 3, 4, 5	lmodscaf 20797	. . . 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 2730	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
9		eqid 2730	. . . . . . 7 ⊢ (dist‘𝐹) = (dist‘𝐹)
10		eqid 2730	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
11		eqid 2730	. . . . . . 7 ⊢ (norm‘𝐹) = (norm‘𝐹)
12		eqid 2730	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
13		eqid 2730	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝐹)‘𝑥) + 1))
14		eqid 2730	. . . . . . 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 766	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ NrmMod)
16		simpr 484	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
17		simplrl 776	. . . . . . 7 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝐹))
18		simplrr 777	. . . . . . 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 24581	. . . . . 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 3126	. . . . 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 776	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐹))
22		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝐹))
23	21, 22	ovresd 7559	. . . . . . . . . . 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 5120	. . . . . . . . . 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 777	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
26		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
27	25, 26	ovresd 7559	. . . . . . . . . . 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 5120	. . . . . . . . . 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 632	. . . . . . . . 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 20794	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥 · 𝑦) = (𝑥( ·_𝑠 ‘𝑊)𝑦))
31	30	ad2antlr 727	. . . . . . . . . . . 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 20794	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧 · 𝑤) = (𝑧( ·_𝑠 ‘𝑊)𝑤))
33	32	adantl 481	. . . . . . . . . . . 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 7408	. . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ NrmMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
36	2, 3, 12, 4	lmodvscl 20791	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·_𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
37	35, 21, 25, 36	syl3anc 1373	. . . . . . . . . . . 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 20791	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝐹) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧( ·_𝑠 ‘𝑊)𝑤) ∈ (Base‘𝑊))
39	35, 22, 26, 38	syl3anc 1373	. . . . . . . . . . . 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 7559	. . . . . . . . . . 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 2765	. . . . . . . . . 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 5120	. . . . . . . . 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 344	. . . . . . . 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 3200	. . . . . . 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 3158	. . . . . 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 3157	. . . . 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 257	. . . 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 3181	. . 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 24575	. . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
50		ngpms 24495	. . . . . 6 ⊢ (𝐹 ∈ NrmGrp → 𝐹 ∈ MetSp)
51	49, 50	syl 17	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ MetSp)
52		msxms 24349	. . . . 5 ⊢ (𝐹 ∈ MetSp → 𝐹 ∈ ∞MetSp)
53		eqid 2730	. . . . . 6 ⊢ ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))) = ((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))
54	4, 53	xmsxmet 24351	. . . . 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 24572	. . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
57		ngpms 24495	. . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
58	56, 57	syl 17	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ MetSp)
59		msxms 24349	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
60		eqid 2730	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
61	2, 60	xmsxmet 24351	. . . . 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 2730	. . . . 5 ⊢ (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹)))) = (MetOpen‘((dist‘𝐹) ↾ ((Base‘𝐹) × (Base‘𝐹))))
64		eqid 2730	. . . . 5 ⊢ (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))
65	63, 64, 64	txmetcn 24443	. . . 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 1373	. . 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 713	. 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 24347	. . . . 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 24347	. . . . 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 7408	. . 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 7408	. 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 2832	1 ⊢ (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 class class class wbr 5110 × cxp 5639 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1c1 11076 + caddc 11078 < clt 11215 / cdiv 11842 2c2 12248 ℝ⁺crp 12958 Basecbs 17186 Scalarcsca 17230 ·_𝑠 cvsca 17231 distcds 17236 TopOpenctopn 17391 LModclmod 20773 ·_sf cscaf 20774 ∞Metcxmet 21256 MetOpencmopn 21261 Cn ccn 23118 ×_t ctx 23454 ∞MetSpcxms 24212 MetSpcms 24213 normcnm 24471 NrmGrpcngp 24472 NrmModcnlm 24475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 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 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-lmod 20775 df-scaf 20776 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cn 23121 df-cnp 23122 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-nm 24477 df-ngp 24478 df-nrg 24480 df-nlm 24481

This theorem is referenced by: nrgtrg 24585 nlmtlm 24589

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