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Theorem ipcn 25293

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 25218	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		eqid 2734	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		ipcn.f	. . . . . 6 ⊢ , = (·_if‘𝑊)
4		eqid 2734	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2734	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	2, 3, 4, 5	phlipf 21687	. . . . 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 25236	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
9	4, 5	clmsscn 25125	. . . . 5 ⊢ (𝑊 ∈ ℂMod → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
10	8, 9	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
11	7, 10	fssd 6753	. . 3 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
12		eqid 2734	. . . . . . 7 ⊢ (·_𝑖‘𝑊) = (·_𝑖‘𝑊)
13		eqid 2734	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
14		eqid 2734	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
15		eqid 2734	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1))
16		eqid 2734	. . . . . . 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 767	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ ℂPreHil)
18		simplrl 777	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝑊))
19		simplrr 778	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (Base‘𝑊))
20		simpr 484	. . . . . . 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 25292	. . . . . 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 3143	. . . . 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 777	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
24		simprl 771	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
25	23, 24	ovresd 7599	. . . . . . . . . . 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 5157	. . . . . . . . . 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 778	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
28		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
29	27, 28	ovresd 7599	. . . . . . . . . . 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 5157	. . . . . . . . . 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 632	. . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
33	32, 23, 27	fovcdmd 7604	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 , 𝑦) ∈ ℂ)
34	32, 24, 28	fovcdmd 7604	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 , 𝑤) ∈ ℂ)
35		eqid 2734	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
36	35	cnmetdval 24806	. . . . . . . . . . . 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 21684	. . . . . . . . . . . . . 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 21684	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧 , 𝑤) = (𝑧(·_𝑖‘𝑊)𝑤))
41	40	adantl 481	. . . . . . . . . . . . 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 7448	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 2774	. . . . . . . . . 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 5157	. . . . . . . . 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 344	. . . . . . . 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 3216	. . . . . . 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 3176	. . . . . 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 3175	. . . . 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 257	. . . 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 3199	. . 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 25220	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
53		ngpms 24628	. . . . . . 7 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
54	52, 53	syl 17	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ MetSp)
55		msxms 24479	. . . . . 6 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
56	54, 55	syl 17	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ∞MetSp)
57		eqid 2734	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
58	2, 57	xmsxmet 24481	. . . . 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 24808	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
61	60	a1i 11	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (abs ∘ − ) ∈ (∞Met‘ℂ))
62		eqid 2734	. . . . 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 24817	. . . . 5 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
65	62, 62, 64	txmetcn 24576	. . . 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 713	. 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 24477	. . . . 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 7448	. . 3 ⊢ (𝑊 ∈ ℂPreHil → (𝐽 ×_t 𝐽) = ((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
72	71	oveq1d 7445	. 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 2841	1 ⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 class class class wbr 5147 × cxp 5686 ↾ cres 5690 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 < clt 11292 − cmin 11489 / cdiv 11917 2c2 12318 ℝ⁺crp 13031 abscabs 15269 Basecbs 17244 Scalarcsca 17300 ·_𝑖cip 17302 distcds 17306 TopOpenctopn 17467 ∞Metcxmet 21366 MetOpencmopn 21371 ℂ_fldccnfld 21381 PreHilcphl 21659 ·_ifcipf 21660 Cn ccn 23247 ×_t ctx 23583 ∞MetSpcxms 24342 MetSpcms 24343 normcnm 24604 NrmGrpcngp 24605 ℂModcclm 25108 ℂPreHilccph 25213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-drng 20747 df-staf 20856 df-srng 20857 df-lmod 20876 df-lmhm 21038 df-lvec 21119 df-sra 21189 df-rgmod 21190 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-phl 21661 df-ipf 21662 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-nm 24610 df-ngp 24611 df-tng 24612 df-nlm 24614 df-clm 25109 df-cph 25215 df-tcph 25216

This theorem is referenced by: cnmpt1ip 25294 cnmpt2ip 25295

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