Theorem cnpwstotbnd 35955

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 2738	. . 3 ⊢ ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})) = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))
2		eqid 2738	. . 3 ⊢ (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
3		eqid 2738	. . 3 ⊢ (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))
4		eqid 2738	. . 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 2738	. . 3 ⊢ (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
6		fvexd 6789	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂfld ↾s 𝐴)) ∈ V)
7		simpr 485	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8		ovex 7308	. . . 4 ⊢ (ℂfld ↾s 𝐴) ∈ V
9		fnconstg 6662	. . . 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 2738	. . 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 23939	. . . . . 6 ⊢ ℂfld ∈ MetSp
13		cnex 10952	. . . . . . . 8 ⊢ ℂ ∈ V
14	13	ssex 5245	. . . . . . 7 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
15	14	ad2antrr 723	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ V)
16		ressms 23682	. . . . . 6 ⊢ ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂfld ↾s 𝐴) ∈ MetSp)
17	12, 15, 16	sylancr 587	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (ℂfld ↾s 𝐴) ∈ MetSp)
18		eqid 2738	. . . . . 6 ⊢ (Base‘(ℂfld ↾s 𝐴)) = (Base‘(ℂfld ↾s 𝐴))
19		eqid 2738	. . . . . 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 23610	. . . . 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 7079	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
23	22	adantl 482	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
24	23	fveq2d 6778	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (dist‘(ℂfld ↾s 𝐴)))
25	23	fveq2d 6778	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘(ℂfld ↾s 𝐴)))
26	25	sqxpeqd 5621	. . . . 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 5892	. . . 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 6778	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘(Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
29	21, 27, 28	3eltr4d 2854	. . 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 35947	. . . . . 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 2738	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝐴) = (ℂfld ↾s 𝐴)
32		cnfldbas 20601	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂfld)
33	31, 32	ressbas2 16949	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
34	33	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
35	34	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
36	21, 35	eleqtrrd 2842	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴))
37		eqid 2738	. . . . . . . . 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 35949	. . . . . . . 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 413	. . . . . . 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 2738	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
43	42	cntotbnd 35954	. . . . . . . 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 3953	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (𝑦 ⊆ 𝐴 ↔ 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴))))
46	45	biimpa 477	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴)))
47		xpss12 5604	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴))))
49	48	resabs1d 5922	. . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ V)
51		cnfldds 20607	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℂfld)
52	31, 51	ressds 17120	. . . . . . . . . . 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 5890	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂfld ↾s 𝐴)) ↾ (𝑦 × 𝑦)))
55	49, 54	eqtr4d 2781	. . . . . . . 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 2823	. . . . . . 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 2823	. . . . . . 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 311	. . . . . 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 413	. . . . 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 381	. . . 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 5890	. . . . 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 2823	. . . 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 2823	. . . 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 311	. . 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 35953	. 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 2738	. . . . . . . 8 ⊢ (Scalar‘(ℂfld ↾s 𝐴)) = (Scalar‘(ℂfld ↾s 𝐴))
69	67, 68	pwsval 17197	. . . . . . 7 ⊢ (((ℂfld ↾s 𝐴) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
70	8, 7, 69	sylancr 587	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
71	70	fveq2d 6778	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))))
72	71	reseq1d 5890	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)))
73	66, 72	eqtrid 2790	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)))
74	73	eleq1d 2823	. 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 2823	. 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 311	1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  {csn 4561   × cxp 5587   ↾ cres 5591   ∘ ccom 5593   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℂcc 10869   − cmin 11205  abscabs 14945  Basecbs 16912   ↾_s cress 16941  Scalarcsca 16965  distcds 16971  X_scprds 17156   ↑_s cpws 17157  Metcmet 20583  ℂ_fldccnfld 20597  MetSpcms 23471  TotBndctotbnd 35924  Bndcbnd 35925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-ec 8500  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-icc 13086  df-fz 13240  df-fl 13512  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-gz 16631  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-topgen 17154  df-prds 17158  df-pws 17160  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-xms 23473  df-ms 23474  df-totbnd 35926  df-bnd 35937

This theorem is referenced by:  rrntotbnd  35994