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Theorem ipcn 25122

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 25047	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		eqid 2729	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		ipcn.f	. . . . . 6 ⊢ , = (·_if‘𝑊)
4		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2729	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	2, 3, 4, 5	phlipf 21537	. . . . 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 25065	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
9	4, 5	clmsscn 24955	. . . . 5 ⊢ (𝑊 ∈ ℂMod → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
10	8, 9	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
11	7, 10	fssd 6687	. . 3 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
12		eqid 2729	. . . . . . 7 ⊢ (·_𝑖‘𝑊) = (·_𝑖‘𝑊)
13		eqid 2729	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
14		eqid 2729	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
15		eqid 2729	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1))
16		eqid 2729	. . . . . . 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 766	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ ℂPreHil)
18		simplrl 776	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝑊))
19		simplrr 777	. . . . . . 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 25121	. . . . . 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 3125	. . . . 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 776	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
24		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
25	23, 24	ovresd 7536	. . . . . . . . . . 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 5112	. . . . . . . . . 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 777	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
28		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
29	27, 28	ovresd 7536	. . . . . . . . . . 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 5112	. . . . . . . . . 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 7541	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 , 𝑦) ∈ ℂ)
34	32, 24, 28	fovcdmd 7541	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 , 𝑤) ∈ ℂ)
35		eqid 2729	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
36	35	cnmetdval 24634	. . . . . . . . . . . 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 21534	. . . . . . . . . . . . . 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 21534	. . . . . . . . . . . . . 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 7387	. . . . . . . . . . . 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 6844	. . . . . . . . . . 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 2764	. . . . . . . . . 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 5112	. . . . . . . . 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 3197	. . . . . . 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 3157	. . . . . 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 3156	. . . . 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 3178	. . 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 25049	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
53		ngpms 24464	. . . . . . 7 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
54	52, 53	syl 17	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ MetSp)
55		msxms 24318	. . . . . 6 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
56	54, 55	syl 17	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ∞MetSp)
57		eqid 2729	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
58	2, 57	xmsxmet 24320	. . . . 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 24636	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
61	60	a1i 11	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (abs ∘ − ) ∈ (∞Met‘ℂ))
62		eqid 2729	. . . . 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 24645	. . . . 5 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
65	62, 62, 64	txmetcn 24412	. . . 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 1373	. . 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 24316	. . . . 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 7387	. . 3 ⊢ (𝑊 ∈ ℂPreHil → (𝐽 ×_t 𝐽) = ((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
72	71	oveq1d 7384	. 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 2831	1 ⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ⊆ wss 3911 class class class wbr 5102 × cxp 5629 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 1c1 11045 + caddc 11047 < clt 11184 − cmin 11381 / cdiv 11811 2c2 12217 ℝ⁺crp 12927 abscabs 15176 Basecbs 17155 Scalarcsca 17199 ·_𝑖cip 17201 distcds 17205 TopOpenctopn 17360 ∞Metcxmet 21225 MetOpencmopn 21230 ℂ_fldccnfld 21240 PreHilcphl 21509 ·_ifcipf 21510 Cn ccn 23087 ×_t ctx 23423 ∞MetSpcxms 24181 MetSpcms 24182 normcnm 24440 NrmGrpcngp 24441 ℂModcclm 24938 ℂPreHilccph 25042

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-mulg 18976 df-subg 19031 df-ghm 19121 df-cntz 19225 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-staf 20724 df-srng 20725 df-lmod 20744 df-lmhm 20905 df-lvec 20986 df-sra 21056 df-rgmod 21057 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-phl 21511 df-ipf 21512 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cn 23090 df-cnp 23091 df-tx 23425 df-hmeo 23618 df-xms 24184 df-ms 24185 df-tms 24186 df-nm 24446 df-ngp 24447 df-tng 24448 df-nlm 24450 df-clm 24939 df-cph 25044 df-tcph 25045

This theorem is referenced by: cnmpt1ip 25123 cnmpt2ip 25124

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