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Theorem cnpwstotbnd 36306
Description: A subset of 𝐴↑𝐼, where 𝐴 βŠ† β„‚, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
cnpwstotbnd.y π‘Œ = ((β„‚fld β†Ύs 𝐴) ↑s 𝐼)
cnpwstotbnd.d 𝐷 = ((distβ€˜π‘Œ) β†Ύ (𝑋 Γ— 𝑋))
Assertion
Ref Expression
cnpwstotbnd ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) ↔ 𝐷 ∈ (Bndβ€˜π‘‹)))

Proof of Theorem cnpwstotbnd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})) = ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))
2 eqid 2733 . . 3 (Baseβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) = (Baseβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
3 eqid 2733 . . 3 (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) = (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))
4 eqid 2733 . . 3 ((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) = ((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))))
5 eqid 2733 . . 3 (distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) = (distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
6 fvexd 6861 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) ∈ V)
7 simpr 486 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ 𝐼 ∈ Fin)
8 ovex 7394 . . . 4 (β„‚fld β†Ύs 𝐴) ∈ V
9 fnconstg 6734 . . . 4 ((β„‚fld β†Ύs 𝐴) ∈ V β†’ (𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}) Fn 𝐼)
108, 9mp1i 13 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}) Fn 𝐼)
11 eqid 2733 . . 3 ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)) = ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋))
12 cnfldms 24162 . . . . . 6 β„‚fld ∈ MetSp
13 cnex 11140 . . . . . . . 8 β„‚ ∈ V
1413ssex 5282 . . . . . . 7 (𝐴 βŠ† β„‚ β†’ 𝐴 ∈ V)
1514ad2antrr 725 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 ∈ V)
16 ressms 23905 . . . . . 6 ((β„‚fld ∈ MetSp ∧ 𝐴 ∈ V) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
1712, 15, 16sylancr 588 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
18 eqid 2733 . . . . . 6 (Baseβ€˜(β„‚fld β†Ύs 𝐴)) = (Baseβ€˜(β„‚fld β†Ύs 𝐴))
19 eqid 2733 . . . . . 6 ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) = ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
2018, 19msmet 23833 . . . . 5 ((β„‚fld β†Ύs 𝐴) ∈ MetSp β†’ ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) ∈ (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
2117, 20syl 17 . . . 4 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) ∈ (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
228fvconst2 7157 . . . . . . 7 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2322adantl 483 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2423fveq2d 6850 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) = (distβ€˜(β„‚fld β†Ύs 𝐴)))
2523fveq2d 6850 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
2625sqxpeqd 5669 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))) = ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
2724, 26reseq12d 5942 . . . 4 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) = ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))))
2825fveq2d 6850 . . . 4 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (Metβ€˜(Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))) = (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
2921, 27, 283eltr4d 2849 . . 3 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) ∈ (Metβ€˜(Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))))
30 totbndbnd 36298 . . . . . 6 ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) β†’ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦))
31 eqid 2733 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝐴) = (β„‚fld β†Ύs 𝐴)
32 cnfldbas 20823 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
3331, 32ressbas2 17128 . . . . . . . . . 10 (𝐴 βŠ† β„‚ β†’ 𝐴 = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
3433ad2antrr 725 . . . . . . . . 9 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
3534fveq2d 6850 . . . . . . . 8 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (Metβ€˜π΄) = (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
3621, 35eleqtrrd 2837 . . . . . . 7 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) ∈ (Metβ€˜π΄))
37 eqid 2733 . . . . . . . . 9 (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) = (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦))
3837bnd2lem 36300 . . . . . . . 8 ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) ∈ (Metβ€˜π΄) ∧ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)) β†’ 𝑦 βŠ† 𝐴)
3938ex 414 . . . . . . 7 (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) ∈ (Metβ€˜π΄) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦) β†’ 𝑦 βŠ† 𝐴))
4036, 39syl 17 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦) β†’ 𝑦 βŠ† 𝐴))
4130, 40syl5 34 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) β†’ 𝑦 βŠ† 𝐴))
42 eqid 2733 . . . . . . . . 9 ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) = ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦))
4342cntotbnd 36305 . . . . . . . 8 (((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦))
4443a1i 11 . . . . . . 7 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ (((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)))
4534sseq2d 3980 . . . . . . . . . . . 12 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 βŠ† 𝐴 ↔ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
4645biimpa 478 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
47 xpss12 5652 . . . . . . . . . . 11 ((𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴)) ∧ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴))) β†’ (𝑦 Γ— 𝑦) βŠ† ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
4846, 46, 47syl2anc 585 . . . . . . . . . 10 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ (𝑦 Γ— 𝑦) βŠ† ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
4948resabs1d 5972 . . . . . . . . 9 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) = ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ (𝑦 Γ— 𝑦)))
5015adantr 482 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝐴 ∈ V)
51 cnfldds 20829 . . . . . . . . . . . 12 (abs ∘ βˆ’ ) = (distβ€˜β„‚fld)
5231, 51ressds 17299 . . . . . . . . . . 11 (𝐴 ∈ V β†’ (abs ∘ βˆ’ ) = (distβ€˜(β„‚fld β†Ύs 𝐴)))
5350, 52syl 17 . . . . . . . . . 10 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ (abs ∘ βˆ’ ) = (distβ€˜(β„‚fld β†Ύs 𝐴)))
5453reseq1d 5940 . . . . . . . . 9 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) = ((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ (𝑦 Γ— 𝑦)))
5549, 54eqtr4d 2776 . . . . . . . 8 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) = ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)))
5655eleq1d 2819 . . . . . . 7 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦)))
5755eleq1d 2819 . . . . . . 7 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦) ↔ ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)))
5844, 56, 573bitr4d 311 . . . . . 6 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)))
5958ex 414 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 βŠ† 𝐴 β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦))))
6041, 40, 59pm5.21ndd 381 . . . 4 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)))
6127reseq1d 5940 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) β†Ύ (𝑦 Γ— 𝑦)) = (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)))
6261eleq1d 2819 . . . 4 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦)))
6361eleq1d 2819 . . . 4 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦) ↔ (((distβ€˜(β„‚fld β†Ύs 𝐴)) β†Ύ ((Baseβ€˜(β„‚fld β†Ύs 𝐴)) Γ— (Baseβ€˜(β„‚fld β†Ύs 𝐴)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)))
6460, 62, 633bitr4d 311 . . 3 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (TotBndβ€˜π‘¦) ↔ (((distβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) β†Ύ ((Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) Γ— (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)))) β†Ύ (𝑦 Γ— 𝑦)) ∈ (Bndβ€˜π‘¦)))
651, 2, 3, 4, 5, 6, 7, 10, 11, 29, 64prdsbnd2 36304 . 2 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)) ∈ (TotBndβ€˜π‘‹) ↔ ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)) ∈ (Bndβ€˜π‘‹)))
66 cnpwstotbnd.d . . . 4 𝐷 = ((distβ€˜π‘Œ) β†Ύ (𝑋 Γ— 𝑋))
67 cnpwstotbnd.y . . . . . . . 8 π‘Œ = ((β„‚fld β†Ύs 𝐴) ↑s 𝐼)
68 eqid 2733 . . . . . . . 8 (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) = (Scalarβ€˜(β„‚fld β†Ύs 𝐴))
6967, 68pwsval 17376 . . . . . . 7 (((β„‚fld β†Ύs 𝐴) ∈ V ∧ 𝐼 ∈ Fin) β†’ π‘Œ = ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
708, 7, 69sylancr 588 . . . . . 6 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ π‘Œ = ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
7170fveq2d 6850 . . . . 5 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (distβ€˜π‘Œ) = (distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))))
7271reseq1d 5940 . . . 4 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ ((distβ€˜π‘Œ) β†Ύ (𝑋 Γ— 𝑋)) = ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)))
7366, 72eqtrid 2785 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ 𝐷 = ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)))
7473eleq1d 2819 . 2 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) ↔ ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)) ∈ (TotBndβ€˜π‘‹)))
7573eleq1d 2819 . 2 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (𝐷 ∈ (Bndβ€˜π‘‹) ↔ ((distβ€˜((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)}))) β†Ύ (𝑋 Γ— 𝑋)) ∈ (Bndβ€˜π‘‹)))
7665, 74, 753bitr4d 311 1 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) ↔ 𝐷 ∈ (Bndβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447   βŠ† wss 3914  {csn 4590   Γ— cxp 5635   β†Ύ cres 5639   ∘ ccom 5641   Fn wfn 6495  β€˜cfv 6500  (class class class)co 7361  Fincfn 8889  β„‚cc 11057   βˆ’ cmin 11393  abscabs 15128  Basecbs 17091   β†Ύs cress 17120  Scalarcsca 17144  distcds 17150  Xscprds 17335   ↑s cpws 17336  Metcmet 20805  β„‚fldccnfld 20819  MetSpcms 23694  TotBndctotbnd 36275  Bndcbnd 36276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-ec 8656  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-icc 13280  df-fz 13434  df-fl 13706  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-gz 16810  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-starv 17156  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-hom 17165  df-cco 17166  df-rest 17312  df-topn 17313  df-topgen 17333  df-prds 17337  df-pws 17339  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-cnfld 20820  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-xms 23696  df-ms 23697  df-totbnd 36277  df-bnd 36288
This theorem is referenced by:  rrntotbnd  36345
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