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Theorem ipcn 25458

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 25383	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		eqid 2765	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		ipcn.f	. . . . . 6 ⊢ , = (·_if‘𝑊)
4		eqid 2765	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2765	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	2, 3, 4, 5	phlipf 21854	. . . . 5 ⊢ (𝑊 ∈ PreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶(Base‘(Scalar‘𝑊)))
7	1, 6	syl 18	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶(Base‘(Scalar‘𝑊)))
8		cphclm 25401	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
9	4, 5	clmsscn 25291	. . . . 5 ⊢ (𝑊 ∈ ℂMod → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
10	8, 9	syl 18	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (Base‘(Scalar‘𝑊)) ⊆ ℂ)
11	7, 10	fssd 6727	. . 3 ⊢ (𝑊 ∈ ℂPreHil → , :((Base‘𝑊) × (Base‘𝑊))⟶ℂ)
12		eqid 2765	. . . . . . 7 ⊢ (·_𝑖‘𝑊) = (·_𝑖‘𝑊)
13		eqid 2765	. . . . . . 7 ⊢ (dist‘𝑊) = (dist‘𝑊)
14		eqid 2765	. . . . . . 7 ⊢ (norm‘𝑊) = (norm‘𝑊)
15		eqid 2765	. . . . . . 7 ⊢ ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1)) = ((𝑟 / 2) / (((norm‘𝑊)‘𝑥) + 1))
16		eqid 2765	. . . . . . 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 779	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑊 ∈ ℂPreHil)
18		simplrl 789	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (Base‘𝑊))
19		simplrr 790	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (Base‘𝑊))
20		simpr 490	. . . . . . 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 25457	. . . . . 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 3159	. . . . 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 789	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
24		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
25	23, 24	ovresd 7586	. . . . . . . . . . 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 5121	. . . . . . . . . 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 790	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
28		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → 𝑤 ∈ (Base‘𝑊))
29	27, 28	ovresd 7586	. . . . . . . . . . 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 5121	. . . . . . . . . 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 644	. . . . . . . . 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 739	. . . . . . . . . . . . 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 7592	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑥 , 𝑦) ∈ ℂ)
34	32, 24, 28	fovcdmd 7592	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊))) → (𝑧 , 𝑤) ∈ ℂ)
35		eqid 2765	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
36	35	cnmetdval 24980	. . . . . . . . . . . 12 ⊢ (((𝑥 , 𝑦) ∈ ℂ ∧ (𝑧 , 𝑤) ∈ ℂ) → ((𝑥 , 𝑦)(abs ∘ − )(𝑧 , 𝑤)) = (abs‘((𝑥 , 𝑦) − (𝑧 , 𝑤))))
37	33, 34, 36	syl2anc 596	. . . . . . . . . . 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 21851	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥 , 𝑦) = (𝑥(·_𝑖‘𝑊)𝑦))
39	23, 27, 38	syl2anc 596	. . . . . . . . . . . . 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 21851	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (Base‘𝑊) ∧ 𝑤 ∈ (Base‘𝑊)) → (𝑧 , 𝑤) = (𝑧(·_𝑖‘𝑊)𝑤))
41	40	adantl 487	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . 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 6889	. . . . . . . . . . 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 2800	. . . . . . . . . 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 5121	. . . . . . . . 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 347	. . . . . . . 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 3229	. . . . . . 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 3191	. . . . . 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 3190	. . . . 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 260	. . . 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 3210	. . 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 25385	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
53		ngpms 24810	. . . . . . 7 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
54	52, 53	syl 18	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ MetSp)
55		msxms 24664	. . . . . 6 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
56	54, 55	syl 18	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ∞MetSp)
57		eqid 2765	. . . . . 6 ⊢ ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))
58	2, 57	xmsxmet 24666	. . . . 5 ⊢ (𝑊 ∈ ∞MetSp → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
59	56, 58	syl 18	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) ∈ (∞Met‘(Base‘𝑊)))
60		cnxmet 24982	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
61	60	a1i 11	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (abs ∘ − ) ∈ (∞Met‘ℂ))
62		eqid 2765	. . . . 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 24991	. . . . 5 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
65	62, 62, 64	txmetcn 24758	. . . 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 1398	. . 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 726	. 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 24662	. . . . 5 ⊢ (𝑊 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
70	54, 69	syl 18	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝐽 = (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))))
71	70, 70	oveq12d 7437	. . 3 ⊢ (𝑊 ∈ ℂPreHil → (𝐽 ×_t 𝐽) = ((MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊)))) ×_t (MetOpen‘((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))))))
72	71	oveq1d 7434	. 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 2868	1 ⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 ⊆ wss 3906 class class class wbr 5111 × cxp 5661 ↾ cres 5665 ∘ ccom 5667 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11115 1c1 11118 + caddc 11120 < clt 11260 − cmin 11458 / cdiv 11888 2c2 12312 ℝ⁺crp 13034 abscabs 15311 Basecbs 17293 Scalarcsca 17337 ·_𝑖cip 17339 distcds 17343 TopOpenctopn 17498 ∞Metcxmet 21559 MetOpencmopn 21564 ℂ_fldccnfld 21574 PreHilcphl 21826 ·_ifcipf 21827 Cn ccn 23433 ×_t ctx 23770 ∞MetSpcxms 24527 MetSpcms 24528 normcnm 24786 NrmGrpcngp 24787 ℂModcclm 25274 ℂPreHilccph 25378

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196 ax-mulf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-pt 17521 df-prds 17524 df-xrs 17580 df-qtop 17585 df-imas 17586 df-xps 17588 df-mre 17662 df-mrc 17663 df-acs 17665 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-mhm 18880 df-submnd 18881 df-grp 19049 df-minusg 19050 df-sbg 19051 df-mulg 19180 df-subg 19235 df-ghm 19330 df-cntz 19433 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-cring 20364 df-oppr 20467 df-dvdsr 20487 df-unit 20488 df-invr 20518 df-dvr 20531 df-rhm 20602 df-subrng 20697 df-subrg 20721 df-drng 20881 df-staf 20994 df-srng 20995 df-lmod 21035 df-lmhm 21195 df-lvec 21276 df-sra 21346 df-rgmod 21347 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-cnfld 21575 df-phl 21828 df-ipf 21829 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-cn 23436 df-cnp 23437 df-tx 23772 df-hmeo 23965 df-xms 24530 df-ms 24531 df-tms 24532 df-nm 24792 df-ngp 24793 df-tng 24794 df-nlm 24796 df-clm 25275 df-cph 25380 df-tcph 25381

This theorem is used by: cnmpt1ip 25459 cnmpt2ip 25460

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