Theorem cnpwstotbnd 34138

Description: A subset of 𝐴↑𝐼, where 𝐴 ⊆ ℂ, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypotheses

Ref	Expression
cnpwstotbnd.y	⊢ 𝑌 = ((ℂfld ↾s 𝐴) ↑s 𝐼)
cnpwstotbnd.d	⊢ 𝐷 = ((dist‘𝑌) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
cnpwstotbnd	⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))

Proof of Theorem cnpwstotbnd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2825	. . 3 ⊢ ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})) = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))
2		eqid 2825	. . 3 ⊢ (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
3		eqid 2825	. . 3 ⊢ (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))
4		eqid 2825	. . 3 ⊢ ((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) = ((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))))
5		eqid 2825	. . 3 ⊢ (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
6		fvexd 6448	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂfld ↾s 𝐴)) ∈ V)
7		simpr 479	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8		ovex 6937	. . . 4 ⊢ (ℂfld ↾s 𝐴) ∈ V
9		fnconstg 6330	. . . 4 ⊢ ((ℂfld ↾s 𝐴) ∈ V → (𝐼 × {(ℂfld ↾s 𝐴)}) Fn 𝐼)
10	8, 9	mp1i 13	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐼 × {(ℂfld ↾s 𝐴)}) Fn 𝐼)
11		eqid 2825	. . 3 ⊢ ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋))
12		cnfldms 22949	. . . . . 6 ⊢ ℂfld ∈ MetSp
13		cnex 10333	. . . . . . . 8 ⊢ ℂ ∈ V
14	13	ssex 5027	. . . . . . 7 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
15	14	ad2antrr 719	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ V)
16		ressms 22701	. . . . . 6 ⊢ ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂfld ↾s 𝐴) ∈ MetSp)
17	12, 15, 16	sylancr 583	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (ℂfld ↾s 𝐴) ∈ MetSp)
18		eqid 2825	. . . . . 6 ⊢ (Base‘(ℂfld ↾s 𝐴)) = (Base‘(ℂfld ↾s 𝐴))
19		eqid 2825	. . . . . 6 ⊢ ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) = ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴))))
20	18, 19	msmet 22632	. . . . 5 ⊢ ((ℂfld ↾s 𝐴) ∈ MetSp → ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘(Base‘(ℂfld ↾s 𝐴))))
21	17, 20	syl 17	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘(Base‘(ℂfld ↾s 𝐴))))
22	8	fvconst2 6725	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
23	22	adantl 475	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
24	23	fveq2d 6437	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (dist‘(ℂfld ↾s 𝐴)))
25	23	fveq2d 6437	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘(ℂfld ↾s 𝐴)))
26	25	sqxpeqd 5374	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))) = ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴))))
27	24, 26	reseq12d 5630	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) = ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))))
28	25	fveq2d 6437	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘(Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
29	21, 27, 28	3eltr4d 2921	. . 3 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) ∈ (Met‘(Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))))
30		totbndbnd 34130	. . . . . 6 ⊢ ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) → (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))
31		eqid 2825	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝐴) = (ℂfld ↾s 𝐴)
32		cnfldbas 20110	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂfld)
33	31, 32	ressbas2 16294	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
34	33	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
35	34	fveq2d 6437	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
36	21, 35	eleqtrrd 2909	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴))
37		eqid 2825	. . . . . . . . 9 ⊢ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦))
38	37	bnd2lem 34132	. . . . . . . 8 ⊢ ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴) ∧ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)) → 𝑦 ⊆ 𝐴)
39	38	ex 403	. . . . . . 7 ⊢ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) → 𝑦 ⊆ 𝐴))
40	36, 39	syl 17	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) → 𝑦 ⊆ 𝐴))
41	30, 40	syl5 34	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) → 𝑦 ⊆ 𝐴))
42		eqid 2825	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
43	42	cntotbnd 34137	. . . . . . . 8 ⊢ (((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))
44	43	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
45	34	sseq2d 3858	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (𝑦 ⊆ 𝐴 ↔ 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴))))
46	45	biimpa 470	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴)))
47		xpss12 5357	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴)) ∧ 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴))) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴))))
48	46, 46, 47	syl2anc 581	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴))))
49	48	resabs1d 5664	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂfld ↾s 𝐴)) ↾ (𝑦 × 𝑦)))
50	15	adantr 474	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ V)
51		cnfldds 20116	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℂfld)
52	31, 51	ressds 16426	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (abs ∘ − ) = (dist‘(ℂfld ↾s 𝐴)))
53	50, 52	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (abs ∘ − ) = (dist‘(ℂfld ↾s 𝐴)))
54	53	reseq1d 5628	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂfld ↾s 𝐴)) ↾ (𝑦 × 𝑦)))
55	49, 54	eqtr4d 2864	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)))
56	55	eleq1d 2891	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦)))
57	55	eleq1d 2891	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
58	44, 56, 57	3bitr4d 303	. . . . . 6 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
59	58	ex 403	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (𝑦 ⊆ 𝐴 → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))))
60	41, 40, 59	pm5.21ndd 371	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
61	27	reseq1d 5628	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)))
62	61	eleq1d 2891	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦)))
63	61	eleq1d 2891	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) ↔ (((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
64	60, 62, 63	3bitr4d 303	. . 3 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
65	1, 2, 3, 4, 5, 6, 7, 10, 11, 29, 64	prdsbnd2 34136	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
66		cnpwstotbnd.d	. . . 4 ⊢ 𝐷 = ((dist‘𝑌) ↾ (𝑋 × 𝑋))
67		cnpwstotbnd.y	. . . . . . . 8 ⊢ 𝑌 = ((ℂfld ↾s 𝐴) ↑s 𝐼)
68		eqid 2825	. . . . . . . 8 ⊢ (Scalar‘(ℂfld ↾s 𝐴)) = (Scalar‘(ℂfld ↾s 𝐴))
69	67, 68	pwsval 16499	. . . . . . 7 ⊢ (((ℂfld ↾s 𝐴) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
70	8, 7, 69	sylancr 583	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
71	70	fveq2d 6437	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))))
72	71	reseq1d 5628	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)))
73	66, 72	syl5eq 2873	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)))
74	73	eleq1d 2891	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋)))
75	73	eleq1d 2891	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (Bnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
76	65, 74, 75	3bitr4d 303	1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ⊆ wss 3798  {csn 4397   × cxp 5340   ↾ cres 5344   ∘ ccom 5346   Fn wfn 6118  ‘cfv 6123  (class class class)co 6905  Fincfn 8222  ℂcc 10250   − cmin 10585  abscabs 14351  Basecbs 16222   ↾_s cress 16223  Scalarcsca 16308  distcds 16314  X_scprds 16459   ↑_s cpws 16460  Metcmet 20092  ℂ_fldccnfld 20106  MetSpcms 22493  TotBndctotbnd 34107  Bndcbnd 34108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-ec 8011  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-q 12072  df-rp 12113  df-xneg 12232  df-xadd 12233  df-xmul 12234  df-icc 12470  df-fz 12620  df-fl 12888  df-seq 13096  df-exp 13155  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-gz 16005  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-starv 16320  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-unif 16328  df-hom 16329  df-cco 16330  df-rest 16436  df-topn 16437  df-topgen 16457  df-prds 16461  df-pws 16463  df-psmet 20098  df-xmet 20099  df-met 20100  df-bl 20101  df-mopn 20102  df-cnfld 20107  df-top 21069  df-topon 21086  df-topsp 21108  df-bases 21121  df-xms 22495  df-ms 22496  df-totbnd 34109  df-bnd 34120

This theorem is referenced by:  rrntotbnd  34177