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Theorem ipcn 25096

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 25021	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		eqid 2724	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		ipcn.f	. . . . . 6 ⊢ , = (·_if‘𝑊)
4		eqid 2724	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2724	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	2, 3, 4, 5	phlipf 21513	. . . . 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 25039	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
9	4, 5	clmsscn 24928	. . . . 5 ⊢ (𝑊 ∈ ℂMod → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
10	8, 9	syl 17	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
11	7, 10	fssd 6725	. . 3 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
12		eqid 2724	. . . . . . 7 ⊢ (·_𝑖‘𝑊) = (·_𝑖‘𝑊)
13		eqid 2724	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
14		eqid 2724	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
15		eqid 2724	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1))
16		eqid 2724	. . . . . . 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 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 25095	. . . . . 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 3138	. . . . 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 7567	. . . . . . . . . . 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 5148	. . . . . . . . . 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 7567	. . . . . . . . . . 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 5148	. . . . . . . . . 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 630	. . . . . . . . 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	fovcdmd 7572	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 , 𝑦) ∈ ℂ)
34	32, 24, 28	fovcdmd 7572	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 , 𝑤) ∈ ℂ)
35		eqid 2724	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
36	35	cnmetdval 24609	. . . . . . . . . . . 12 ⊢ (((𝑥 , 𝑦) ∈ ℂ ∧ (𝑧 , 𝑤) ∈ ℂ) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) = (abs‘((𝑥 , 𝑦) − (𝑧 , 𝑤))))
37	33, 34, 36	syl2anc 583	. . . . . . . . . . 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 21510	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥 , 𝑦) = (𝑥(·_𝑖‘𝑊)𝑦))
39	23, 27, 38	syl2anc 583	. . . . . . . . . . . . 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 21510	. . . . . . . . . . . . . 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 7419	. . . . . . . . . . . 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 6885	. . . . . . . . . . 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 5148	. . . . . . . . 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 3208	. . . . . . 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 3170	. . . . . 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 3169	. . . . 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 3192	. . 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 25023	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
53		ngpms 24431	. . . . . . 7 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
54	52, 53	syl 17	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ MetSp)
55		msxms 24282	. . . . . 6 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
56	54, 55	syl 17	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ∞MetSp)
57		eqid 2724	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
58	2, 57	xmsxmet 24284	. . . . 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 24611	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
61	60	a1i 11	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (abs ∘ − ) ∈ (∞Met‘ℂ))
62		eqid 2724	. . . . 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 24620	. . . . 5 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
65	62, 62, 64	txmetcn 24379	. . . 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 1368	. . 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 24280	. . . . 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 7419	. . 3 ⊢ (𝑊 ∈ ℂPreHil → (𝐽 ×_t 𝐽) = ((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
72	71	oveq1d 7416	. 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 2828	1 ⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 class class class wbr 5138 × cxp 5664 ↾ cres 5668 ∘ ccom 5670 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 1c1 11107 + caddc 11109 < clt 11245 − cmin 11441 / cdiv 11868 2c2 12264 ℝ⁺crp 12971 abscabs 15178 Basecbs 17143 Scalarcsca 17199 ·_𝑖cip 17201 distcds 17205 TopOpenctopn 17366 ∞Metcxmet 21213 MetOpencmopn 21218 ℂ_fldccnfld 21228 PreHilcphl 21485 ·_ifcipf 21486 Cn ccn 23050 ×_t ctx 23386 ∞MetSpcxms 24145 MetSpcms 24146 normcnm 24407 NrmGrpcngp 24408 ℂModcclm 24911 ℂPreHilccph 25016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 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 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-drng 20579 df-staf 20678 df-srng 20679 df-lmod 20698 df-lmhm 20860 df-lvec 20941 df-sra 21011 df-rgmod 21012 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-cnfld 21229 df-phl 21487 df-ipf 21488 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cn 23053 df-cnp 23054 df-tx 23388 df-hmeo 23581 df-xms 24148 df-ms 24149 df-tms 24150 df-nm 24413 df-ngp 24414 df-tng 24415 df-nlm 24417 df-clm 24912 df-cph 25018 df-tcph 25019

This theorem is referenced by: cnmpt1ip 25097 cnmpt2ip 25098

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