Theorem cnpwstotbnd 37784

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 2729	. . 3 ⊢ ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})) = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))
2		eqid 2729	. . 3 ⊢ (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
3		eqid 2729	. . 3 ⊢ (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))
4		eqid 2729	. . 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 2729	. . 3 ⊢ (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
6		fvexd 6855	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂfld ↾s 𝐴)) ∈ V)
7		simpr 484	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8		ovex 7402	. . . 4 ⊢ (ℂfld ↾s 𝐴) ∈ V
9		fnconstg 6730	. . . 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 2729	. . 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 24696	. . . . . 6 ⊢ ℂfld ∈ MetSp
13		cnex 11125	. . . . . . . 8 ⊢ ℂ ∈ V
14	13	ssex 5271	. . . . . . 7 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
15	14	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ V)
16		ressms 24447	. . . . . 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 2729	. . . . . 6 ⊢ (Base‘(ℂfld ↾s 𝐴)) = (Base‘(ℂfld ↾s 𝐴))
19		eqid 2729	. . . . . 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 24378	. . . . 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 7160	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
23	22	adantl 481	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
24	23	fveq2d 6844	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (dist‘(ℂfld ↾s 𝐴)))
25	23	fveq2d 6844	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘(ℂfld ↾s 𝐴)))
26	25	sqxpeqd 5663	. . . . 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 5940	. . . 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 6844	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘(Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
29	21, 27, 28	3eltr4d 2843	. . 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 37776	. . . . . 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 2729	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝐴) = (ℂfld ↾s 𝐴)
32		cnfldbas 21300	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂfld)
33	31, 32	ressbas2 17184	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
34	33	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
35	34	fveq2d 6844	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
36	21, 35	eleqtrrd 2831	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴))
37		eqid 2729	. . . . . . . . 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 37778	. . . . . . . 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 412	. . . . . . 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 2729	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
43	42	cntotbnd 37783	. . . . . . . 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 3976	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (𝑦 ⊆ 𝐴 ↔ 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴))))
46	45	biimpa 476	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴)))
47		xpss12 5646	. . . . . . . . . . 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 5968	. . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ V)
51		cnfldds 21308	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℂfld)
52	31, 51	ressds 17349	. . . . . . . . . . 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 5938	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂfld ↾s 𝐴)) ↾ (𝑦 × 𝑦)))
55	49, 54	eqtr4d 2767	. . . . . . . 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 2813	. . . . . . 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 2813	. . . . . . 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 412	. . . . 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 379	. . . 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 5938	. . . . 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 2813	. . . 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 2813	. . . 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 37782	. 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 2729	. . . . . . . 8 ⊢ (Scalar‘(ℂfld ↾s 𝐴)) = (Scalar‘(ℂfld ↾s 𝐴))
69	67, 68	pwsval 17425	. . . . . . 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 6844	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))))
72	71	reseq1d 5938	. . . 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 2776	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)))
74	73	eleq1d 2813	. 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 2813	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  {csn 4585   × cxp 5629   ↾ cres 5633   ∘ ccom 5635   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  ℂcc 11042   − cmin 11381  abscabs 15176  Basecbs 17155   ↾_s cress 17176  Scalarcsca 17199  distcds 17205  X_scprds 17384   ↑_s cpws 17385  Metcmet 21282  ℂ_fldccnfld 21296  MetSpcms 24239  TotBndctotbnd 37753  Bndcbnd 37754

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

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-iun 4953  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-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-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-ec 8650  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  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-icc 13289  df-fz 13445  df-fl 13730  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-gz 16877  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-topgen 17382  df-prds 17386  df-pws 17388  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-cnfld 21297  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-xms 24241  df-ms 24242  df-totbnd 37755  df-bnd 37766

This theorem is referenced by:  rrntotbnd  37823