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Theorem ipcn 24410

Description: The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)

**Hypotheses**
Ref	Expression
ipcn.f	⊢ , = (·_if‘𝑊)
ipcn.j	⊢ 𝐽 = (TopOpen‘𝑊)
ipcn.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
ipcn	⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

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

Step	Hyp	Ref	Expression
1		cphphl 24335	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		eqid 2738	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		ipcn.f	. . . . . 6 ⊢ , = (·_if‘𝑊)
4		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2738	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	2, 3, 4, 5	phlipf 20857	. . . . 5 ⊢ (𝑊 ∈ PreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶(Base‘(Scalar‘𝑊)))
7	1, 6	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶(Base‘(Scalar‘𝑊)))
8		cphclm 24353	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
9	4, 5	clmsscn 24242	. . . . 5 ⊢ (𝑊 ∈ ℂMod → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
10	8, 9	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
11	7, 10	fssd 6618	. . 3 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
12		eqid 2738	. . . . . . 7 ⊢ (·_𝑖‘𝑊) = (·_𝑖‘𝑊)
13		eqid 2738	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
14		eqid 2738	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
15		eqid 2738	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1))
16		eqid 2738	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑦) + ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1)))) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑦) + ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1))))
17		simpll 764	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ ℂPreHil)
18		simplrl 774	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝑊))
19		simplrr 775	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (Base‘𝑊))
20		simpr 485	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
21	2, 12, 13, 14, 15, 16, 17, 18, 19, 20	ipcnlem1 24409	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟))
22	21	ralrimiva 3103	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟))
23		simplrl 774	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
24		simprl 768	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
25	23, 24	ovresd 7439	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) = (𝑥(dist‘𝑊)𝑧))
26	25	breq1d 5084	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ↔ (𝑥(dist‘𝑊)𝑧) < 𝑠))
27		simplrr 775	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
28		simprr 770	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
29	27, 28	ovresd 7439	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) = (𝑦(dist‘𝑊)𝑤))
30	29	breq1d 5084	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠 ↔ (𝑦(dist‘𝑊)𝑤) < 𝑠))
31	26, 30	anbi12d 631	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) ↔ ((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠)))
32	11	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
33	32, 23, 27	fovrnd 7444	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 , 𝑦) ∈ ℂ)
34	32, 24, 28	fovrnd 7444	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 , 𝑤) ∈ ℂ)
35		eqid 2738	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
36	35	cnmetdval 23934	. . . . . . . . . . . 12 ⊢ (((𝑥 , 𝑦) ∈ ℂ ∧ (𝑧 , 𝑤) ∈ ℂ) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) = (abs‘((𝑥 , 𝑦) − (𝑧 , 𝑤))))
37	33, 34, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) = (abs‘((𝑥 , 𝑦) − (𝑧 , 𝑤))))
38	2, 12, 3	ipfval 20854	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥 , 𝑦) = (𝑥(·_𝑖‘𝑊)𝑦))
39	23, 27, 38	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 , 𝑦) = (𝑥(·_𝑖‘𝑊)𝑦))
40	2, 12, 3	ipfval 20854	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧 , 𝑤) = (𝑧(·_𝑖‘𝑊)𝑤))
41	40	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 , 𝑤) = (𝑧(·_𝑖‘𝑊)𝑤))
42	39, 41	oveq12d 7293	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥 , 𝑦) − (𝑧 , 𝑤)) = ((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤)))
43	42	fveq2d 6778	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (abs‘((𝑥 , 𝑦) − (𝑧 , 𝑤))) = (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))))
44	37, 43	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) = (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))))
45	44	breq1d 5084	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟 ↔ (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟))
46	31, 45	imbi12d 345	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → ((((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟) ↔ (((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟)))
47	46	2ralbidva 3128	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟) ↔ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟)))
48	47	rexbidv 3226	. . . . . 6 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟) ↔ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟)))
49	48	ralbidv 3112	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟) ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥(dist‘𝑊)𝑧) < 𝑠 ∧ (𝑦(dist‘𝑊)𝑤) < 𝑠) → (abs‘((𝑥(·_𝑖‘𝑊)𝑦) − (𝑧(·_𝑖‘𝑊)𝑤))) < 𝑟)))
50	22, 49	mpbird 256	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟))
51	50	ralrimivva 3123	. . 3 ⊢ (𝑊 ∈ ℂPreHil → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟))
52		cphngp 24337	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
53		ngpms 23756	. . . . . . 7 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
54	52, 53	syl 17	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ MetSp)
55		msxms 23607	. . . . . 6 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
56	54, 55	syl 17	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ∞MetSp)
57		eqid 2738	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
58	2, 57	xmsxmet 23609	. . . . 5 ⊢ (𝑊 ∈ ∞MetSp → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
59	56, 58	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
60		cnxmet 23936	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
61	60	a1i 11	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (abs ∘ − ) ∈ (∞Met‘ℂ))
62		eqid 2738	. . . . 5 ⊢ (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))
63		ipcn.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
64	63	cnfldtopn 23945	. . . . 5 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
65	62, 62, 64	txmetcn 23704	. . . 4 ⊢ ((((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)) ∧ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) → ( , ∈ (((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn 𝐾) ↔ ( , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ ∧ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟))))
66	59, 59, 61, 65	syl3anc 1370	. . 3 ⊢ (𝑊 ∈ ℂPreHil → ( , ∈ (((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn 𝐾) ↔ ( , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ ∧ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑥((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑧) < 𝑠 ∧ (𝑦((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))𝑤) < 𝑠) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) < 𝑟))))
67	11, 51, 66	mpbir2and 710	. 2 ⊢ (𝑊 ∈ ℂPreHil → , ∈ (((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn 𝐾))
68		ipcn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
69	68, 2, 57	mstopn 23605	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
70	54, 69	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
71	70, 70	oveq12d 7293	. . 3 ⊢ (𝑊 ∈ ℂPreHil → (𝐽 ×_t 𝐽) = ((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
72	71	oveq1d 7290	. 2 ⊢ (𝑊 ∈ ℂPreHil → ((𝐽 ×_t 𝐽) Cn 𝐾) = (((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))) Cn 𝐾))
73	67, 72	eleqtrrd 2842	1 ⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 × cxp 5587 ↾ cres 5591 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 1c1 10872 + caddc 10874 < clt 11009 − cmin 11205 / cdiv 11632 2c2 12028 ℝ⁺crp 12730 abscabs 14945 Basecbs 16912 Scalarcsca 16965 ·_𝑖cip 16967 distcds 16971 TopOpenctopn 17132 ∞Metcxmet 20582 MetOpencmopn 20587 ℂ_fldccnfld 20597 PreHilcphl 20829 ·_ifcipf 20830 Cn ccn 22375 ×_t ctx 22711 ∞MetSpcxms 23470 MetSpcms 23471 normcnm 23732 NrmGrpcngp 23733 ℂModcclm 24225 ℂPreHilccph 24330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lmhm 20284 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-phl 20831 df-ipf 20832 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cn 22378 df-cnp 22379 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-tng 23740 df-nlm 23742 df-clm 24226 df-cph 24332 df-tcph 24333

This theorem is referenced by: cnmpt1ip 24411 cnmpt2ip 24412

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