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Theorem cnpwstotbnd 37178
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 2726 . . 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 2726 . . 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 2726 . . 3 (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) = (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))
4 eqid 2726 . . 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 2726 . . 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 6900 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) ∈ V)
7 simpr 484 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ 𝐼 ∈ Fin)
8 ovex 7438 . . . 4 (β„‚fld β†Ύs 𝐴) ∈ V
9 fnconstg 6773 . . . 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 2726 . . 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 24647 . . . . . 6 β„‚fld ∈ MetSp
13 cnex 11193 . . . . . . . 8 β„‚ ∈ V
1413ssex 5314 . . . . . . 7 (𝐴 βŠ† β„‚ β†’ 𝐴 ∈ V)
1514ad2antrr 723 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 ∈ V)
16 ressms 24390 . . . . . 6 ((β„‚fld ∈ MetSp ∧ 𝐴 ∈ V) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
1712, 15, 16sylancr 586 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
18 eqid 2726 . . . . . 6 (Baseβ€˜(β„‚fld β†Ύs 𝐴)) = (Baseβ€˜(β„‚fld β†Ύs 𝐴))
19 eqid 2726 . . . . . 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 24318 . . . . 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 7201 . . . . . . 7 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2322adantl 481 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2423fveq2d 6889 . . . . 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 6889 . . . . . 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 5701 . . . . 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 5976 . . . 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 6889 . . . 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 2842 . . 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 37170 . . . . . 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 2726 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝐴) = (β„‚fld β†Ύs 𝐴)
32 cnfldbas 21244 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
3331, 32ressbas2 17191 . . . . . . . . . 10 (𝐴 βŠ† β„‚ β†’ 𝐴 = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
3433ad2antrr 723 . . . . . . . . 9 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 = (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
3534fveq2d 6889 . . . . . . . 8 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (Metβ€˜π΄) = (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
3621, 35eleqtrrd 2830 . . . . . . 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 2726 . . . . . . . . 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 37172 . . . . . . . 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 412 . . . . . . 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 2726 . . . . . . . . 9 ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) = ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦))
4342cntotbnd 37177 . . . . . . . 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 4009 . . . . . . . . . . . 12 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 βŠ† 𝐴 ↔ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴))))
4645biimpa 476 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
47 xpss12 5684 . . . . . . . . . . 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 583 . . . . . . . . . 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 6006 . . . . . . . . 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 480 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝐴 ∈ V)
51 cnfldds 21252 . . . . . . . . . . . 12 (abs ∘ βˆ’ ) = (distβ€˜β„‚fld)
5231, 51ressds 17364 . . . . . . . . . . 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 5974 . . . . . . . . 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 2769 . . . . . . . 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 2812 . . . . . . 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 2812 . . . . . . 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 412 . . . . 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 379 . . . 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 5974 . . . . 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 2812 . . . 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 2812 . . . 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 37176 . 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 2726 . . . . . . . 8 (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) = (Scalarβ€˜(β„‚fld β†Ύs 𝐴))
6967, 68pwsval 17441 . . . . . . 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 586 . . . . . 6 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ π‘Œ = ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
7170fveq2d 6889 . . . . 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 5974 . . . 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 2778 . . 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 2812 . 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 2812 . 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943  {csn 4623   Γ— cxp 5667   β†Ύ cres 5671   ∘ ccom 5673   Fn wfn 6532  β€˜cfv 6537  (class class class)co 7405  Fincfn 8941  β„‚cc 11110   βˆ’ cmin 11448  abscabs 15187  Basecbs 17153   β†Ύs cress 17182  Scalarcsca 17209  distcds 17215  Xscprds 17400   ↑s cpws 17401  Metcmet 21226  β„‚fldccnfld 21240  MetSpcms 24179  TotBndctotbnd 37147  Bndcbnd 37148
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-ec 8707  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-icc 13337  df-fz 13491  df-fl 13763  df-seq 13973  df-exp 14033  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-gz 16872  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-rest 17377  df-topn 17378  df-topgen 17398  df-prds 17402  df-pws 17404  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-cnfld 21241  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-xms 24181  df-ms 24182  df-totbnd 37149  df-bnd 37160
This theorem is referenced by:  rrntotbnd  37217
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