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Theorem cnpwstotbnd 37804
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 2737 . . 3 ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})) = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))
2 eqid 2737 . . 3 (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
3 eqid 2737 . . 3 (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))
4 eqid 2737 . . 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 2737 . . 3 (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
6 fvexd 6921 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂflds 𝐴)) ∈ V)
7 simpr 484 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8 ovex 7464 . . . 4 (ℂflds 𝐴) ∈ V
9 fnconstg 6796 . . . 4 ((ℂflds 𝐴) ∈ V → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
108, 9mp1i 13 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
11 eqid 2737 . . 3 ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋))
12 cnfldms 24796 . . . . . 6 fld ∈ MetSp
13 cnex 11236 . . . . . . . 8 ℂ ∈ V
1413ssex 5321 . . . . . . 7 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
1514ad2antrr 726 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 ∈ V)
16 ressms 24539 . . . . . 6 ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂflds 𝐴) ∈ MetSp)
1712, 15, 16sylancr 587 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (ℂflds 𝐴) ∈ MetSp)
18 eqid 2737 . . . . . 6 (Base‘(ℂflds 𝐴)) = (Base‘(ℂflds 𝐴))
19 eqid 2737 . . . . . 6 ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) = ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2018, 19msmet 24467 . . . . 5 ((ℂflds 𝐴) ∈ MetSp → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘(Base‘(ℂflds 𝐴))))
2117, 20syl 17 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘(Base‘(ℂflds 𝐴))))
228fvconst2 7224 . . . . . . 7 (𝑥𝐼 → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2322adantl 481 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2423fveq2d 6910 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (dist‘(ℂflds 𝐴)))
2523fveq2d 6910 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘(ℂflds 𝐴)))
2625sqxpeqd 5717 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2724, 26reseq12d 5998 . . . 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)))))
2825fveq2d 6910 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘(Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂflds 𝐴))))
2921, 27, 283eltr4d 2856 . . 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))))
30 totbndbnd 37796 . . . . . 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))
31 eqid 2737 . . . . . . . . . . 11 (ℂflds 𝐴) = (ℂflds 𝐴)
32 cnfldbas 21368 . . . . . . . . . . 11 ℂ = (Base‘ℂfld)
3331, 32ressbas2 17283 . . . . . . . . . 10 (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂflds 𝐴)))
3433ad2antrr 726 . . . . . . . . 9 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 = (Base‘(ℂflds 𝐴)))
3534fveq2d 6910 . . . . . . . 8 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂflds 𝐴))))
3621, 35eleqtrrd 2844 . . . . . . 7 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴))
37 eqid 2737 . . . . . . . . 9 (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦))
3837bnd2lem 37798 . . . . . . . 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)
3938ex 412 . . . . . . 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))
4036, 39syl 17 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) → 𝑦𝐴))
4130, 40syl5 34 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) → 𝑦𝐴))
42 eqid 2737 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
4342cntotbnd 37803 . . . . . . . 8 (((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))
4443a1i 11 . . . . . . 7 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ ((abs ∘ − ) ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
4534sseq2d 4016 . . . . . . . . . . . 12 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (𝑦𝐴𝑦 ⊆ (Base‘(ℂflds 𝐴))))
4645biimpa 476 . . . . . . . . . . 11 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → 𝑦 ⊆ (Base‘(ℂflds 𝐴)))
47 xpss12 5700 . . . . . . . . . . 11 ((𝑦 ⊆ (Base‘(ℂflds 𝐴)) ∧ 𝑦 ⊆ (Base‘(ℂflds 𝐴))) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
4846, 46, 47syl2anc 584 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (𝑦 × 𝑦) ⊆ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
4948resabs1d 6026 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5015adantr 480 . . . . . . . . . . 11 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → 𝐴 ∈ V)
51 cnfldds 21376 . . . . . . . . . . . 12 (abs ∘ − ) = (dist‘ℂfld)
5231, 51ressds 17454 . . . . . . . . . . 11 (𝐴 ∈ V → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5350, 52syl 17 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5453reseq1d 5996 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5549, 54eqtr4d 2780 . . . . . . . 8 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)))
5655eleq1d 2826 . . . . . . 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)))
5755eleq1d 2826 . . . . . . 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)))
5844, 56, 573bitr4d 311 . . . . . 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)))
5958ex 412 . . . . 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))))
6041, 40, 59pm5.21ndd 379 . . . 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)))
6127reseq1d 5996 . . . . 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)))
6261eleq1d 2826 . . . 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)))
6361eleq1d 2826 . . . 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)))
6460, 62, 633bitr4d 311 . . 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)))
651, 2, 3, 4, 5, 6, 7, 10, 11, 29, 64prdsbnd2 37802 . 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)))
66 cnpwstotbnd.d . . . 4 𝐷 = ((dist‘𝑌) ↾ (𝑋 × 𝑋))
67 cnpwstotbnd.y . . . . . . . 8 𝑌 = ((ℂflds 𝐴) ↑s 𝐼)
68 eqid 2737 . . . . . . . 8 (Scalar‘(ℂflds 𝐴)) = (Scalar‘(ℂflds 𝐴))
6967, 68pwsval 17531 . . . . . . 7 (((ℂflds 𝐴) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
708, 7, 69sylancr 587 . . . . . 6 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
7170fveq2d 6910 . . . . 5 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))))
7271reseq1d 5996 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7366, 72eqtrid 2789 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7473eleq1d 2826 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋)))
7573eleq1d 2826 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (Bnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
7665, 74, 753bitr4d 311 1 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  {csn 4626   × cxp 5683  cres 5687  ccom 5689   Fn wfn 6556  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  cmin 11492  abscabs 15273  Basecbs 17247  s cress 17274  Scalarcsca 17300  distcds 17306  Xscprds 17490  s cpws 17491  Metcmet 21350  fldccnfld 21364  MetSpcms 24328  TotBndctotbnd 37773  Bndcbnd 37774
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-fz 13548  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-gz 16968  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-topgen 17488  df-prds 17492  df-pws 17494  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-xms 24330  df-ms 24331  df-totbnd 37775  df-bnd 37786
This theorem is referenced by:  rrntotbnd  37843
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