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Theorem cnpwstotbnd 37311
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 2728 . . 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 2728 . . 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 2728 . . 3 (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) = (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))
4 eqid 2728 . . 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 2728 . . 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 6917 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) ∈ V)
7 simpr 483 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ 𝐼 ∈ Fin)
8 ovex 7459 . . . 4 (β„‚fld β†Ύs 𝐴) ∈ V
9 fnconstg 6790 . . . 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 2728 . . 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 24720 . . . . . 6 β„‚fld ∈ MetSp
13 cnex 11229 . . . . . . . 8 β„‚ ∈ V
1413ssex 5325 . . . . . . 7 (𝐴 βŠ† β„‚ β†’ 𝐴 ∈ V)
1514ad2antrr 724 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 ∈ V)
16 ressms 24463 . . . . . 6 ((β„‚fld ∈ MetSp ∧ 𝐴 ∈ V) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
1712, 15, 16sylancr 585 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
18 eqid 2728 . . . . . 6 (Baseβ€˜(β„‚fld β†Ύs 𝐴)) = (Baseβ€˜(β„‚fld β†Ύs 𝐴))
19 eqid 2728 . . . . . 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 24391 . . . . 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 7222 . . . . . . 7 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2322adantl 480 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2423fveq2d 6906 . . . . 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 6906 . . . . . 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 5714 . . . . 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 5990 . . . 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 6906 . . . 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 2844 . . 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 37303 . . . . . 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 2728 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝐴) = (β„‚fld β†Ύs 𝐴)
32 cnfldbas 21297 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
3331, 32ressbas2 17227 . . . . . . . . . 10 (𝐴 βŠ† β„‚ β†’ 𝐴 = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
3433ad2antrr 724 . . . . . . . . 9 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
3534fveq2d 6906 . . . . . . . 8 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (Metβ€˜π΄) = (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
3621, 35eleqtrrd 2832 . . . . . . 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 2728 . . . . . . . . 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 37305 . . . . . . . 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 411 . . . . . . 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 2728 . . . . . . . . 9 ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) = ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦))
4342cntotbnd 37310 . . . . . . . 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 4014 . . . . . . . . . . . 12 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 βŠ† 𝐴 ↔ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
4645biimpa 475 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
47 xpss12 5697 . . . . . . . . . . 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 582 . . . . . . . . . 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 6017 . . . . . . . . 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 479 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝐴 ∈ V)
51 cnfldds 21305 . . . . . . . . . . . 12 (abs ∘ βˆ’ ) = (distβ€˜β„‚fld)
5231, 51ressds 17400 . . . . . . . . . . 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 5988 . . . . . . . . 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 2771 . . . . . . . 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 2814 . . . . . . 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 2814 . . . . . . 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 310 . . . . . 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 411 . . . . 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 378 . . . 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 5988 . . . . 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 2814 . . . 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 2814 . . . 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 310 . . 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 37309 . 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 2728 . . . . . . . 8 (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) = (Scalarβ€˜(β„‚fld β†Ύs 𝐴))
6967, 68pwsval 17477 . . . . . . 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 585 . . . . . 6 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ π‘Œ = ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
7170fveq2d 6906 . . . . 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 5988 . . . 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 2780 . . 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 2814 . 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 2814 . 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 310 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βŠ† wss 3949  {csn 4632   Γ— cxp 5680   β†Ύ cres 5684   ∘ ccom 5686   Fn wfn 6548  β€˜cfv 6553  (class class class)co 7426  Fincfn 8972  β„‚cc 11146   βˆ’ cmin 11484  abscabs 15223  Basecbs 17189   β†Ύs cress 17218  Scalarcsca 17245  distcds 17251  Xscprds 17436   ↑s cpws 17437  Metcmet 21279  β„‚fldccnfld 21293  MetSpcms 24252  TotBndctotbnd 37280  Bndcbnd 37281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225  ax-pre-sup 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-ec 8735  df-map 8855  df-pm 8856  df-ixp 8925  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-sup 9475  df-inf 9476  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12253  df-2 12315  df-3 12316  df-4 12317  df-5 12318  df-6 12319  df-7 12320  df-8 12321  df-9 12322  df-n0 12513  df-z 12599  df-dec 12718  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13134  df-xadd 13135  df-xmul 13136  df-icc 13373  df-fz 13527  df-fl 13799  df-seq 14009  df-exp 14069  df-cj 15088  df-re 15089  df-im 15090  df-sqrt 15224  df-abs 15225  df-gz 16908  df-struct 17125  df-sets 17142  df-slot 17160  df-ndx 17172  df-base 17190  df-ress 17219  df-plusg 17255  df-mulr 17256  df-starv 17257  df-sca 17258  df-vsca 17259  df-ip 17260  df-tset 17261  df-ple 17262  df-ds 17264  df-unif 17265  df-hom 17266  df-cco 17267  df-rest 17413  df-topn 17414  df-topgen 17434  df-prds 17438  df-pws 17440  df-psmet 21285  df-xmet 21286  df-met 21287  df-bl 21288  df-mopn 21289  df-cnfld 21294  df-top 22824  df-topon 22841  df-topsp 22863  df-bases 22877  df-xms 24254  df-ms 24255  df-totbnd 37282  df-bnd 37293
This theorem is referenced by:  rrntotbnd  37350
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