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Theorem cnpwstotbnd 36256
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 2736 . . 3 ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})) = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))
2 eqid 2736 . . 3 (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
3 eqid 2736 . . 3 (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))
4 eqid 2736 . . 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 2736 . . 3 (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
6 fvexd 6857 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂflds 𝐴)) ∈ V)
7 simpr 485 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8 ovex 7390 . . . 4 (ℂflds 𝐴) ∈ V
9 fnconstg 6730 . . . 4 ((ℂflds 𝐴) ∈ V → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
108, 9mp1i 13 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
11 eqid 2736 . . 3 ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋))
12 cnfldms 24139 . . . . . 6 fld ∈ MetSp
13 cnex 11132 . . . . . . . 8 ℂ ∈ V
1413ssex 5278 . . . . . . 7 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
1514ad2antrr 724 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 ∈ V)
16 ressms 23882 . . . . . 6 ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂflds 𝐴) ∈ MetSp)
1712, 15, 16sylancr 587 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (ℂflds 𝐴) ∈ MetSp)
18 eqid 2736 . . . . . 6 (Base‘(ℂflds 𝐴)) = (Base‘(ℂflds 𝐴))
19 eqid 2736 . . . . . 6 ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) = ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2018, 19msmet 23810 . . . . 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 7153 . . . . . . 7 (𝑥𝐼 → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2322adantl 482 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2423fveq2d 6846 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (dist‘(ℂflds 𝐴)))
2523fveq2d 6846 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘(ℂflds 𝐴)))
2625sqxpeqd 5665 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2724, 26reseq12d 5938 . . . 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 6846 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘(Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂflds 𝐴))))
2921, 27, 283eltr4d 2853 . . 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 36248 . . . . . 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 2736 . . . . . . . . . . 11 (ℂflds 𝐴) = (ℂflds 𝐴)
32 cnfldbas 20800 . . . . . . . . . . 11 ℂ = (Base‘ℂfld)
3331, 32ressbas2 17120 . . . . . . . . . 10 (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂflds 𝐴)))
3433ad2antrr 724 . . . . . . . . 9 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 = (Base‘(ℂflds 𝐴)))
3534fveq2d 6846 . . . . . . . 8 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂflds 𝐴))))
3621, 35eleqtrrd 2841 . . . . . . 7 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴))
37 eqid 2736 . . . . . . . . 9 (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦))
3837bnd2lem 36250 . . . . . . . 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 413 . . . . . . 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 2736 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
4342cntotbnd 36255 . . . . . . . 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 3976 . . . . . . . . . . . 12 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (𝑦𝐴𝑦 ⊆ (Base‘(ℂflds 𝐴))))
4645biimpa 477 . . . . . . . . . . 11 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → 𝑦 ⊆ (Base‘(ℂflds 𝐴)))
47 xpss12 5648 . . . . . . . . . . 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 5968 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5015adantr 481 . . . . . . . . . . 11 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → 𝐴 ∈ V)
51 cnfldds 20806 . . . . . . . . . . . 12 (abs ∘ − ) = (dist‘ℂfld)
5231, 51ressds 17291 . . . . . . . . . . 11 (𝐴 ∈ V → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5350, 52syl 17 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5453reseq1d 5936 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5549, 54eqtr4d 2779 . . . . . . . 8 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)))
5655eleq1d 2822 . . . . . . 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 2822 . . . . . . 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 310 . . . . . 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 413 . . . . 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 380 . . . 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 5936 . . . . 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 2822 . . . 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 2822 . . . 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 310 . . 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 36254 . 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 2736 . . . . . . . 8 (Scalar‘(ℂflds 𝐴)) = (Scalar‘(ℂflds 𝐴))
6967, 68pwsval 17368 . . . . . . 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 6846 . . . . 5 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))))
7271reseq1d 5936 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7366, 72eqtrid 2788 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7473eleq1d 2822 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋)))
7573eleq1d 2822 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (Bnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
7665, 74, 753bitr4d 310 1 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  wss 3910  {csn 4586   × cxp 5631  cres 5635  ccom 5637   Fn wfn 6491  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  cmin 11385  abscabs 15119  Basecbs 17083  s cress 17112  Scalarcsca 17136  distcds 17142  Xscprds 17327  s cpws 17328  Metcmet 20782  fldccnfld 20796  MetSpcms 23671  TotBndctotbnd 36225  Bndcbnd 36226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-fz 13425  df-fl 13697  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-gz 16802  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-topgen 17325  df-prds 17329  df-pws 17331  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-xms 23673  df-ms 23674  df-totbnd 36227  df-bnd 36238
This theorem is referenced by:  rrntotbnd  36295
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