Theorem cnpwstotbnd 35090

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 2821	. . 3 ⊢ ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})) = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))
2		eqid 2821	. . 3 ⊢ (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
3		eqid 2821	. . 3 ⊢ (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))
4		eqid 2821	. . 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 2821	. . 3 ⊢ (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
6		fvexd 6685	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂfld ↾s 𝐴)) ∈ V)
7		simpr 487	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8		ovex 7189	. . . 4 ⊢ (ℂfld ↾s 𝐴) ∈ V
9		fnconstg 6567	. . . 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 2821	. . 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 23384	. . . . . 6 ⊢ ℂfld ∈ MetSp
13		cnex 10618	. . . . . . . 8 ⊢ ℂ ∈ V
14	13	ssex 5225	. . . . . . 7 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
15	14	ad2antrr 724	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ V)
16		ressms 23136	. . . . . 6 ⊢ ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂfld ↾s 𝐴) ∈ MetSp)
17	12, 15, 16	sylancr 589	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (ℂfld ↾s 𝐴) ∈ MetSp)
18		eqid 2821	. . . . . 6 ⊢ (Base‘(ℂfld ↾s 𝐴)) = (Base‘(ℂfld ↾s 𝐴))
19		eqid 2821	. . . . . 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 23067	. . . . 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 6966	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
23	22	adantl 484	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥) = (ℂfld ↾s 𝐴))
24	23	fveq2d 6674	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (dist‘(ℂfld ↾s 𝐴)))
25	23	fveq2d 6674	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘(ℂfld ↾s 𝐴)))
26	25	sqxpeqd 5587	. . . . 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 5854	. . . 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 6674	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘(Base‘((𝐼 × {(ℂfld ↾s 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
29	21, 27, 28	3eltr4d 2928	. . 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 35082	. . . . . 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 2821	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝐴) = (ℂfld ↾s 𝐴)
32		cnfldbas 20549	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂfld)
33	31, 32	ressbas2 16555	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
34	33	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (Base‘(ℂfld ↾s 𝐴)))
35	34	fveq2d 6674	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂfld ↾s 𝐴))))
36	21, 35	eleqtrrd 2916	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘(ℂfld ↾s 𝐴)) ↾ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴))
37		eqid 2821	. . . . . . . . 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 35084	. . . . . . . 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 415	. . . . . . 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 2821	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
43	42	cntotbnd 35089	. . . . . . . 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 3999	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (𝑦 ⊆ 𝐴 ↔ 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴))))
46	45	biimpa 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ (Base‘(ℂfld ↾s 𝐴)))
47		xpss12 5570	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂfld ↾s 𝐴)) × (Base‘(ℂfld ↾s 𝐴))))
49	48	resabs1d 5884	. . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ V)
51		cnfldds 20555	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℂfld)
52	31, 51	ressds 16686	. . . . . . . . . . 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 5852	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂfld ↾s 𝐴)) ↾ (𝑦 × 𝑦)))
55	49, 54	eqtr4d 2859	. . . . . . . 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 2897	. . . . . . 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 2897	. . . . . . 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 313	. . . . . 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 415	. . . . 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 383	. . . 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 5852	. . . . 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 2897	. . . 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 2897	. . . 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 313	. . 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 35088	. 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 2821	. . . . . . . 8 ⊢ (Scalar‘(ℂfld ↾s 𝐴)) = (Scalar‘(ℂfld ↾s 𝐴))
69	67, 68	pwsval 16759	. . . . . . 7 ⊢ (((ℂfld ↾s 𝐴) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
70	8, 7, 69	sylancr 589	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)})))
71	70	fveq2d 6674	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))))
72	71	reseq1d 5852	. . . 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 2868	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld ↾s 𝐴)}))) ↾ (𝑋 × 𝑋)))
74	73	eleq1d 2897	. 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 2897	. 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 313	1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  {csn 4567   × cxp 5553   ↾ cres 5557   ∘ ccom 5559   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156  Fincfn 8509  ℂcc 10535   − cmin 10870  abscabs 14593  Basecbs 16483   ↾_s cress 16484  Scalarcsca 16568  distcds 16574  X_scprds 16719   ↑_s cpws 16720  Metcmet 20531  ℂ_fldccnfld 20545  MetSpcms 22928  TotBndctotbnd 35059  Bndcbnd 35060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-ec 8291  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-icc 12746  df-fz 12894  df-fl 13163  df-seq 13371  df-exp 13431  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-gz 16266  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-topgen 16717  df-prds 16721  df-pws 16723  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-xms 22930  df-ms 22931  df-totbnd 35061  df-bnd 35072

This theorem is referenced by:  rrntotbnd  35129