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Theorem cnpwstotbnd 37784
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 2735 . . 3 ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})) = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))
2 eqid 2735 . . 3 (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
3 eqid 2735 . . 3 (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))
4 eqid 2735 . . 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 2735 . . 3 (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
6 fvexd 6922 . . 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 6797 . . . 4 ((ℂflds 𝐴) ∈ V → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
108, 9mp1i 13 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
11 eqid 2735 . . 3 ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋))
12 cnfldms 24812 . . . . . 6 fld ∈ MetSp
13 cnex 11234 . . . . . . . 8 ℂ ∈ V
1413ssex 5327 . . . . . . 7 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
1514ad2antrr 726 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 ∈ V)
16 ressms 24555 . . . . . 6 ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂflds 𝐴) ∈ MetSp)
1712, 15, 16sylancr 587 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (ℂflds 𝐴) ∈ MetSp)
18 eqid 2735 . . . . . 6 (Base‘(ℂflds 𝐴)) = (Base‘(ℂflds 𝐴))
19 eqid 2735 . . . . . 6 ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) = ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2018, 19msmet 24483 . . . . 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 6911 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (dist‘(ℂflds 𝐴)))
2523fveq2d 6911 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘(ℂflds 𝐴)))
2625sqxpeqd 5721 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2724, 26reseq12d 6001 . . . 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 6911 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘(Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂflds 𝐴))))
2921, 27, 283eltr4d 2854 . . 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 37776 . . . . . 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 2735 . . . . . . . . . . 11 (ℂflds 𝐴) = (ℂflds 𝐴)
32 cnfldbas 21386 . . . . . . . . . . 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 6911 . . . . . . . 8 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂflds 𝐴))))
3621, 35eleqtrrd 2842 . . . . . . 7 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴))
37 eqid 2735 . . . . . . . . 9 (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦))
3837bnd2lem 37778 . . . . . . . 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 2735 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
4342cntotbnd 37783 . . . . . . . 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 4028 . . . . . . . . . . . 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 5704 . . . . . . . . . . 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 6028 . . . . . . . . 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 21394 . . . . . . . . . . . 12 (abs ∘ − ) = (dist‘ℂfld)
5231, 51ressds 17456 . . . . . . . . . . 11 (𝐴 ∈ V → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5350, 52syl 17 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5453reseq1d 5999 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5549, 54eqtr4d 2778 . . . . . . . 8 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)))
5655eleq1d 2824 . . . . . . 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 2824 . . . . . . 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 5999 . . . . 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 2824 . . . 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 2824 . . . 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 37782 . 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 2735 . . . . . . . 8 (Scalar‘(ℂflds 𝐴)) = (Scalar‘(ℂflds 𝐴))
6967, 68pwsval 17533 . . . . . . 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 6911 . . . . 5 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))))
7271reseq1d 5999 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7366, 72eqtrid 2787 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7473eleq1d 2824 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋)))
7573eleq1d 2824 . 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 1537  wcel 2106  Vcvv 3478  wss 3963  {csn 4631   × cxp 5687  cres 5691  ccom 5693   Fn wfn 6558  cfv 6563  (class class class)co 7431  Fincfn 8984  cc 11151  cmin 11490  abscabs 15270  Basecbs 17245  s cress 17274  Scalarcsca 17301  distcds 17307  Xscprds 17492  s cpws 17493  Metcmet 21368  fldccnfld 21382  MetSpcms 24344  TotBndctotbnd 37753  Bndcbnd 37754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-icc 13391  df-fz 13545  df-fl 13829  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-gz 16964  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-topgen 17490  df-prds 17494  df-pws 17496  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-xms 24346  df-ms 24347  df-totbnd 37755  df-bnd 37766
This theorem is referenced by:  rrntotbnd  37823
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