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Theorem cnpwstotbnd 32590
Description: A subset of 𝐴𝐼, where 𝐴 ⊆ ℂ, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
cnpwstotbnd.y 𝑌 = ((ℂflds 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.
StepHypRef Expression
1 eqid 2609 . . 3 ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})) = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))
2 eqid 2609 . . 3 (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
3 eqid 2609 . . 3 (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))
4 eqid 2609 . . 3 ((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) = ((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))))
5 eqid 2609 . . 3 (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
6 fvex 6098 . . . 4 (Scalar‘(ℂflds 𝐴)) ∈ V
76a1i 11 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂflds 𝐴)) ∈ V)
8 simpr 475 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
9 ovex 6555 . . . 4 (ℂflds 𝐴) ∈ V
10 fnconstg 5991 . . . 4 ((ℂflds 𝐴) ∈ V → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
119, 10mp1i 13 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
12 eqid 2609 . . 3 ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋))
13 cnfldms 22337 . . . . . 6 fld ∈ MetSp
14 cnex 9874 . . . . . . . 8 ℂ ∈ V
1514ssex 4725 . . . . . . 7 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
1615ad2antrr 757 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 ∈ V)
17 ressms 22089 . . . . . 6 ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂflds 𝐴) ∈ MetSp)
1813, 16, 17sylancr 693 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (ℂflds 𝐴) ∈ MetSp)
19 eqid 2609 . . . . . 6 (Base‘(ℂflds 𝐴)) = (Base‘(ℂflds 𝐴))
20 eqid 2609 . . . . . 6 ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) = ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2119, 20msmet 22020 . . . . 5 ((ℂflds 𝐴) ∈ MetSp → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘(Base‘(ℂflds 𝐴))))
2218, 21syl 17 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘(Base‘(ℂflds 𝐴))))
239fvconst2 6352 . . . . . . 7 (𝑥𝐼 → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2423adantl 480 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2524fveq2d 6092 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (dist‘(ℂflds 𝐴)))
2624fveq2d 6092 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘(ℂflds 𝐴)))
2726sqxpeqd 5055 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2825, 27reseq12d 5305 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) = ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))))
2926fveq2d 6092 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘(Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂflds 𝐴))))
3022, 28, 293eltr4d 2702 . . 3 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) ∈ (Met‘(Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))))
31 totbndbnd 32582 . . . . . 6 ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))
32 eqid 2609 . . . . . . . . . . 11 (ℂflds 𝐴) = (ℂflds 𝐴)
33 cnfldbas 19520 . . . . . . . . . . 11 ℂ = (Base‘ℂfld)
3432, 33ressbas2 15707 . . . . . . . . . 10 (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂflds 𝐴)))
3534ad2antrr 757 . . . . . . . . 9 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 = (Base‘(ℂflds 𝐴)))
3635fveq2d 6092 . . . . . . . 8 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂflds 𝐴))))
3722, 36eleqtrrd 2690 . . . . . . 7 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴))
38 eqid 2609 . . . . . . . . 9 (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦))
3938bnd2lem 32584 . . . . . . . 8 ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴) ∧ (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)) → 𝑦𝐴)
4039ex 448 . . . . . . 7 (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) → 𝑦𝐴))
4137, 40syl 17 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) → 𝑦𝐴))
4231, 41syl5 33 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) → 𝑦𝐴))
43 eqid 2609 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
4443cntotbnd 32589 . . . . . . . 8 (((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))
4544a1i 11 . . . . . . 7 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
4635sseq2d 3595 . . . . . . . . . . . 12 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (𝑦𝐴𝑦 ⊆ (Base‘(ℂflds 𝐴))))
4746biimpa 499 . . . . . . . . . . 11 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → 𝑦 ⊆ (Base‘(ℂflds 𝐴)))
48 xpss12 5137 . . . . . . . . . . 11 ((𝑦 ⊆ (Base‘(ℂflds 𝐴)) ∧ 𝑦 ⊆ (Base‘(ℂflds 𝐴))) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
4947, 47, 48syl2anc 690 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
5049resabs1d 5335 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5116adantr 479 . . . . . . . . . . 11 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → 𝐴 ∈ V)
52 cnfldds 19526 . . . . . . . . . . . 12 (abs ∘ − ) = (dist‘ℂfld)
5332, 52ressds 15845 . . . . . . . . . . 11 (𝐴 ∈ V → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5451, 53syl 17 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5554reseq1d 5303 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5650, 55eqtr4d 2646 . . . . . . . 8 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)))
5756eleq1d 2671 . . . . . . 7 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦)))
5856eleq1d 2671 . . . . . . 7 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
5945, 57, 583bitr4d 298 . . . . . 6 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
6059ex 448 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (𝑦𝐴 → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))))
6142, 41, 60pm5.21ndd 367 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
6228reseq1d 5303 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)))
6362eleq1d 2671 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦)))
6462eleq1d 2671 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) ↔ (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
6561, 63, 643bitr4d 298 . . 3 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (((dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) ↾ ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
661, 2, 3, 4, 5, 7, 8, 11, 12, 30, 65prdsbnd2 32588 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
67 cnpwstotbnd.d . . . 4 𝐷 = ((dist‘𝑌) ↾ (𝑋 × 𝑋))
68 cnpwstotbnd.y . . . . . . . 8 𝑌 = ((ℂflds 𝐴) ↑s 𝐼)
69 eqid 2609 . . . . . . . 8 (Scalar‘(ℂflds 𝐴)) = (Scalar‘(ℂflds 𝐴))
7068, 69pwsval 15918 . . . . . . 7 (((ℂflds 𝐴) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
719, 8, 70sylancr 693 . . . . . 6 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
7271fveq2d 6092 . . . . 5 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))))
7372reseq1d 5303 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7467, 73syl5eq 2655 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7574eleq1d 2671 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋)))
7674eleq1d 2671 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (Bnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
7766, 75, 763bitr4d 298 1 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  {csn 4124   × cxp 5026  cres 5030  ccom 5032   Fn wfn 5785  cfv 5790  (class class class)co 6527  Fincfn 7819  cc 9791  cmin 10118  abscabs 13771  Basecbs 15644  s cress 15645  Scalarcsca 15720  distcds 15726  Xscprds 15878  s cpws 15879  Metcme 19502  fldccnfld 19516  MetSpcmt 21881  TotBndctotbnd 32559  Bndcbnd 32560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-ec 7609  df-map 7724  df-pm 7725  df-ixp 7773  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-icc 12012  df-fz 12156  df-fl 12413  df-seq 12622  df-exp 12681  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-gz 15421  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-mulr 15731  df-starv 15732  df-sca 15733  df-vsca 15734  df-ip 15735  df-tset 15736  df-ple 15737  df-ds 15740  df-unif 15741  df-hom 15742  df-cco 15743  df-rest 15855  df-topn 15856  df-topgen 15876  df-prds 15880  df-pws 15882  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-cnfld 19517  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-xms 21883  df-ms 21884  df-totbnd 32561  df-bnd 32572
This theorem is referenced by:  rrntotbnd  32629
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