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Theorem cnpwstotbnd 36660
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 2732 . . 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 2732 . . 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 2732 . . 3 (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯)) = (Baseβ€˜((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯))
4 eqid 2732 . . 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 2732 . . 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 6906 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) ∈ V)
7 simpr 485 . . 3 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ 𝐼 ∈ Fin)
8 ovex 7441 . . . 4 (β„‚fld β†Ύs 𝐴) ∈ V
9 fnconstg 6779 . . . 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 2732 . . 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 24291 . . . . . 6 β„‚fld ∈ MetSp
13 cnex 11190 . . . . . . . 8 β„‚ ∈ V
1413ssex 5321 . . . . . . 7 (𝐴 βŠ† β„‚ β†’ 𝐴 ∈ V)
1514ad2antrr 724 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ 𝐴 ∈ V)
16 ressms 24034 . . . . . 6 ((β„‚fld ∈ MetSp ∧ 𝐴 ∈ V) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
1712, 15, 16sylancr 587 . . . . 5 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (β„‚fld β†Ύs 𝐴) ∈ MetSp)
18 eqid 2732 . . . . . 6 (Baseβ€˜(β„‚fld β†Ύs 𝐴)) = (Baseβ€˜(β„‚fld β†Ύs 𝐴))
19 eqid 2732 . . . . . 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 23962 . . . . 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 7204 . . . . . . 7 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2322adantl 482 . . . . . 6 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ ((𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})β€˜π‘₯) = (β„‚fld β†Ύs 𝐴))
2423fveq2d 6895 . . . . 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 6895 . . . . . 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 5708 . . . . 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 5982 . . . 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 6895 . . . 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 2848 . . 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 36652 . . . . . 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 2732 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝐴) = (β„‚fld β†Ύs 𝐴)
32 cnfldbas 20947 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
3331, 32ressbas2 17181 . . . . . . . . . 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 6895 . . . . . . . 8 (((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) β†’ (Metβ€˜π΄) = (Metβ€˜(Baseβ€˜(β„‚fld β†Ύs 𝐴))))
3621, 35eleqtrrd 2836 . . . . . . 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 2732 . . . . . . . . 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 36654 . . . . . . . 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 413 . . . . . . 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 2732 . . . . . . . . 9 ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦)) = ((abs ∘ βˆ’ ) β†Ύ (𝑦 Γ— 𝑦))
4342cntotbnd 36659 . . . . . . . 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 477 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝐴)))
47 xpss12 5691 . . . . . . . . . . 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 584 . . . . . . . . . 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 6012 . . . . . . . . 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 481 . . . . . . . . . . 11 ((((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 βŠ† 𝐴) β†’ 𝐴 ∈ V)
51 cnfldds 20953 . . . . . . . . . . . 12 (abs ∘ βˆ’ ) = (distβ€˜β„‚fld)
5231, 51ressds 17354 . . . . . . . . . . 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 5980 . . . . . . . . 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 2775 . . . . . . . 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 2818 . . . . . . 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 2818 . . . . . . 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 413 . . . . 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 380 . . . 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 5980 . . . . 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 2818 . . . 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 2818 . . . 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 36658 . 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 2732 . . . . . . . 8 (Scalarβ€˜(β„‚fld β†Ύs 𝐴)) = (Scalarβ€˜(β„‚fld β†Ύs 𝐴))
6967, 68pwsval 17431 . . . . . . 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 587 . . . . . 6 ((𝐴 βŠ† β„‚ ∧ 𝐼 ∈ Fin) β†’ π‘Œ = ((Scalarβ€˜(β„‚fld β†Ύs 𝐴))Xs(𝐼 Γ— {(β„‚fld β†Ύs 𝐴)})))
7170fveq2d 6895 . . . . 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 5980 . . . 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 2784 . . 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 2818 . 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 2818 . 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  {csn 4628   Γ— cxp 5674   β†Ύ cres 5678   ∘ ccom 5680   Fn wfn 6538  β€˜cfv 6543  (class class class)co 7408  Fincfn 8938  β„‚cc 11107   βˆ’ cmin 11443  abscabs 15180  Basecbs 17143   β†Ύs cress 17172  Scalarcsca 17199  distcds 17205  Xscprds 17390   ↑s cpws 17391  Metcmet 20929  β„‚fldccnfld 20943  MetSpcms 23823  TotBndctotbnd 36629  Bndcbnd 36630
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-ec 8704  df-map 8821  df-pm 8822  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-q 12932  df-rp 12974  df-xneg 13091  df-xadd 13092  df-xmul 13093  df-icc 13330  df-fz 13484  df-fl 13756  df-seq 13966  df-exp 14027  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-gz 16862  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17367  df-topn 17368  df-topgen 17388  df-prds 17392  df-pws 17394  df-psmet 20935  df-xmet 20936  df-met 20937  df-bl 20938  df-mopn 20939  df-cnfld 20944  df-top 22395  df-topon 22412  df-topsp 22434  df-bases 22448  df-xms 23825  df-ms 23826  df-totbnd 36631  df-bnd 36642
This theorem is referenced by:  rrntotbnd  36699
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