Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnpwstotbnd Structured version   Visualization version   GIF version

Theorem cnpwstotbnd 34956
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 2818 . . 3 ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})) = ((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))
2 eqid 2818 . . 3 (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (Base‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
3 eqid 2818 . . 3 (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))
4 eqid 2818 . . 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 2818 . . 3 (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)})))
6 fvexd 6678 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (Scalar‘(ℂflds 𝐴)) ∈ V)
7 simpr 485 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin)
8 ovex 7178 . . . 4 (ℂflds 𝐴) ∈ V
9 fnconstg 6560 . . . 4 ((ℂflds 𝐴) ∈ V → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
108, 9mp1i 13 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐼 × {(ℂflds 𝐴)}) Fn 𝐼)
11 eqid 2818 . . 3 ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋))
12 cnfldms 23311 . . . . . 6 fld ∈ MetSp
13 cnex 10606 . . . . . . . 8 ℂ ∈ V
1413ssex 5216 . . . . . . 7 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
1514ad2antrr 722 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 ∈ V)
16 ressms 23063 . . . . . 6 ((ℂfld ∈ MetSp ∧ 𝐴 ∈ V) → (ℂflds 𝐴) ∈ MetSp)
1712, 15, 16sylancr 587 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (ℂflds 𝐴) ∈ MetSp)
18 eqid 2818 . . . . . 6 (Base‘(ℂflds 𝐴)) = (Base‘(ℂflds 𝐴))
19 eqid 2818 . . . . . 6 ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) = ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2018, 19msmet 22994 . . . . 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 6958 . . . . . . 7 (𝑥𝐼 → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2322adantl 482 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((𝐼 × {(ℂflds 𝐴)})‘𝑥) = (ℂflds 𝐴))
2423fveq2d 6667 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (dist‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (dist‘(ℂflds 𝐴)))
2523fveq2d 6667 . . . . . 6 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) = (Base‘(ℂflds 𝐴)))
2625sqxpeqd 5580 . . . . 5 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥)) × (Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴))))
2724, 26reseq12d 5847 . . . 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 6667 . . . 4 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘(Base‘((𝐼 × {(ℂflds 𝐴)})‘𝑥))) = (Met‘(Base‘(ℂflds 𝐴))))
2921, 27, 283eltr4d 2925 . . 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 34948 . . . . . 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 2818 . . . . . . . . . . 11 (ℂflds 𝐴) = (ℂflds 𝐴)
32 cnfldbas 20477 . . . . . . . . . . 11 ℂ = (Base‘ℂfld)
3331, 32ressbas2 16543 . . . . . . . . . 10 (𝐴 ⊆ ℂ → 𝐴 = (Base‘(ℂflds 𝐴)))
3433ad2antrr 722 . . . . . . . . 9 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → 𝐴 = (Base‘(ℂflds 𝐴)))
3534fveq2d 6667 . . . . . . . 8 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → (Met‘𝐴) = (Met‘(Base‘(ℂflds 𝐴))))
3621, 35eleqtrrd 2913 . . . . . . 7 (((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) → ((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ∈ (Met‘𝐴))
37 eqid 2818 . . . . . . . . 9 (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦))
3837bnd2lem 34950 . . . . . . . 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 2818 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦))
4342cntotbnd 34955 . . . . . . . 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 3996 . . . . . . . . . . . 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 5563 . . . . . . . . . . 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 5877 . . . . . . . . 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 20483 . . . . . . . . . . . 12 (abs ∘ − ) = (dist‘ℂfld)
5231, 51ressds 16674 . . . . . . . . . . 11 (𝐴 ∈ V → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5350, 52syl 17 . . . . . . . . . 10 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (abs ∘ − ) = (dist‘(ℂflds 𝐴)))
5453reseq1d 5845 . . . . . . . . 9 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → ((abs ∘ − ) ↾ (𝑦 × 𝑦)) = ((dist‘(ℂflds 𝐴)) ↾ (𝑦 × 𝑦)))
5549, 54eqtr4d 2856 . . . . . . . 8 ((((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) ∧ 𝑥𝐼) ∧ 𝑦𝐴) → (((dist‘(ℂflds 𝐴)) ↾ ((Base‘(ℂflds 𝐴)) × (Base‘(ℂflds 𝐴)))) ↾ (𝑦 × 𝑦)) = ((abs ∘ − ) ↾ (𝑦 × 𝑦)))
5655eleq1d 2894 . . . . . . 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 2894 . . . . . . 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 312 . . . . . 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 381 . . . 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 5845 . . . . 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 2894 . . . 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 2894 . . . 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 312 . . 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 34954 . 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 2818 . . . . . . . 8 (Scalar‘(ℂflds 𝐴)) = (Scalar‘(ℂflds 𝐴))
6967, 68pwsval 16747 . . . . . . 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 6667 . . . . 5 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (dist‘𝑌) = (dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))))
7271reseq1d 5845 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → ((dist‘𝑌) ↾ (𝑋 × 𝑋)) = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7366, 72syl5eq 2865 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → 𝐷 = ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)))
7473eleq1d 2894 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (TotBnd‘𝑋)))
7573eleq1d 2894 . 2 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (Bnd‘𝑋) ↔ ((dist‘((Scalar‘(ℂflds 𝐴))Xs(𝐼 × {(ℂflds 𝐴)}))) ↾ (𝑋 × 𝑋)) ∈ (Bnd‘𝑋)))
7665, 74, 753bitr4d 312 1 ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  {csn 4557   × cxp 5546  cres 5550  ccom 5552   Fn wfn 6343  cfv 6348  (class class class)co 7145  Fincfn 8497  cc 10523  cmin 10858  abscabs 14581  Basecbs 16471  s cress 16472  Scalarcsca 16556  distcds 16562  Xscprds 16707  s cpws 16708  Metcmet 20459  fldccnfld 20473  MetSpcms 22855  TotBndctotbnd 34925  Bndcbnd 34926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-ec 8280  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-icc 12733  df-fz 12881  df-fl 13150  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-gz 16254  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-topgen 16705  df-prds 16709  df-pws 16711  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-xms 22857  df-ms 22858  df-totbnd 34927  df-bnd 34938
This theorem is referenced by:  rrntotbnd  34995
  Copyright terms: Public domain W3C validator