| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2737 | . . 3
⊢
((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)})) =
((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)})) | 
| 2 |  | eqid 2737 | . . 3
⊢
(Base‘((Scalar‘(ℂfld ↾s
𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)})))
= (Base‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)}))) | 
| 3 |  | eqid 2737 | . . 3
⊢
(Base‘((𝐼
× {(ℂfld ↾s 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂfld
↾s 𝐴)})‘𝑥)) | 
| 4 |  | eqid 2737 | . . 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 2737 | . . 3
⊢
(dist‘((Scalar‘(ℂfld ↾s
𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)})))
= (dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)}))) | 
| 6 |  | fvexd 6921 | . . 3
⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) →
(Scalar‘(ℂfld ↾s 𝐴)) ∈ V) | 
| 7 |  | simpr 484 | . . 3
⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin) | 
| 8 |  | ovex 7464 | . . . 4
⊢
(ℂfld ↾s 𝐴) ∈ V | 
| 9 |  | fnconstg 6796 | . . . 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 2737 | . . 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 24796 | . . . . . 6
⊢
ℂfld ∈ MetSp | 
| 13 |  | cnex 11236 | . . . . . . . 8
⊢ ℂ
∈ V | 
| 14 | 13 | ssex 5321 | . . . . . . 7
⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V) | 
| 15 | 14 | ad2antrr 726 | . . . . . 6
⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ V) | 
| 16 |  | ressms 24539 | . . . . . 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 2737 | . . . . . 6
⊢
(Base‘(ℂfld ↾s 𝐴)) = (Base‘(ℂfld
↾s 𝐴)) | 
| 19 |  | eqid 2737 | . . . . . 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 24467 | . . . . 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 7224 | . . . . . . 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 6910 | . . . . 5
⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝐼 × {(ℂfld
↾s 𝐴)})‘𝑥)) = (dist‘(ℂfld
↾s 𝐴))) | 
| 25 | 23 | fveq2d 6910 | . . . . . 6
⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {(ℂfld
↾s 𝐴)})‘𝑥)) = (Base‘(ℂfld
↾s 𝐴))) | 
| 26 | 25 | sqxpeqd 5717 | . . . . 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 5998 | . . . 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 6910 | . . . 4
⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘(Base‘((𝐼 × {(ℂfld
↾s 𝐴)})‘𝑥))) =
(Met‘(Base‘(ℂfld ↾s 𝐴)))) | 
| 29 | 21, 27, 28 | 3eltr4d 2856 | . . 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 37796 | . . . . . 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 2737 | . . . . . . . . . . 11
⊢
(ℂfld ↾s 𝐴) = (ℂfld
↾s 𝐴) | 
| 32 |  | cnfldbas 21368 | . . . . . . . . . . 11
⊢ ℂ =
(Base‘ℂfld) | 
| 33 | 31, 32 | ressbas2 17283 | . . . . . . . . . 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 6910 | . . . . . . . 8
⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → (Met‘𝐴) =
(Met‘(Base‘(ℂfld ↾s 𝐴)))) | 
| 36 | 21, 35 | eleqtrrd 2844 | . . . . . . 7
⊢ (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) → ((dist‘(ℂfld
↾s 𝐴))
↾ ((Base‘(ℂfld ↾s 𝐴)) ×
(Base‘(ℂfld ↾s 𝐴)))) ∈ (Met‘𝐴)) | 
| 37 |  | eqid 2737 | . . . . . . . . 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 37798 | . . . . . . . 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 2737 | . . . . . . . . 9
⊢ ((abs
∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)) | 
| 43 | 42 | cntotbnd 37803 | . . . . . . . 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 4016 | . . . . . . . . . . . 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 5700 | . . . . . . . . . . 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 6026 | . . . . . . . . 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 21376 | . . . . . . . . . . . 12
⊢ (abs
∘ − ) = (dist‘ℂfld) | 
| 52 | 31, 51 | ressds 17454 | . . . . . . . . . . 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 5996 | . . . . . . . . 9
⊢ ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ⊆ 𝐴) → ((abs ∘ − ) ↾
(𝑦 × 𝑦)) =
((dist‘(ℂfld ↾s 𝐴)) ↾ (𝑦 × 𝑦))) | 
| 55 | 49, 54 | eqtr4d 2780 | . . . . . . . 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 2826 | . . . . . . 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 2826 | . . . . . . 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 5996 | . . . . 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 2826 | . . . 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 2826 | . . . 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 37802 | . 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 2737 | . . . . . . . 8
⊢
(Scalar‘(ℂfld ↾s 𝐴)) =
(Scalar‘(ℂfld ↾s 𝐴)) | 
| 69 | 67, 68 | pwsval 17531 | . . . . . . 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 6910 | . . . . 5
⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) →
(dist‘𝑌) =
(dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)})))) | 
| 72 | 71 | reseq1d 5996 | . . . 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 2789 | . . 3
⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 =
((dist‘((Scalar‘(ℂfld ↾s 𝐴))Xs(𝐼 × {(ℂfld
↾s 𝐴)})))
↾ (𝑋 × 𝑋))) | 
| 74 | 73 | eleq1d 2826 | . 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 2826 | . 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))) |