Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1ffval Structured version   Visualization version   GIF version

Theorem hdmap1ffval 39397
 Description: Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.)
Hypothesis
Ref Expression
hdmap1val.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
hdmap1ffval (𝐾𝑋 → (HDMap1‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))
Distinct variable groups:   𝑤,𝐻   𝑎,𝑐,𝑑,𝑗,𝑚,𝑛,𝑢,𝑣,𝑤,𝐾   ,𝑎,𝑥,𝑐,𝑑,𝑗,𝑚,𝑛,𝑢,𝑣,𝑤
Allowed substitution hints:   𝐻(𝑥,𝑣,𝑢,,𝑗,𝑚,𝑛,𝑎,𝑐,𝑑)   𝐾(𝑥,)   𝑋(𝑥,𝑤,𝑣,𝑢,,𝑗,𝑚,𝑛,𝑎,𝑐,𝑑)

Proof of Theorem hdmap1ffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3428 . 2 (𝐾𝑋𝐾 ∈ V)
2 fveq2 6662 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 hdmap1val.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2811 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6662 . . . . . . 7 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6664 . . . . . 6 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
7 fveq2 6662 . . . . . . . . . 10 (𝑘 = 𝐾 → (LCDual‘𝑘) = (LCDual‘𝐾))
87fveq1d 6664 . . . . . . . . 9 (𝑘 = 𝐾 → ((LCDual‘𝑘)‘𝑤) = ((LCDual‘𝐾)‘𝑤))
9 fveq2 6662 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (mapd‘𝑘) = (mapd‘𝐾))
109fveq1d 6664 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((mapd‘𝑘)‘𝑤) = ((mapd‘𝐾)‘𝑤))
1110sbceq1d 3703 . . . . . . . . . . 11 (𝑘 = 𝐾 → ([((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
1211sbcbidv 3753 . . . . . . . . . 10 (𝑘 = 𝐾 → ([(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
1312sbcbidv 3753 . . . . . . . . 9 (𝑘 = 𝐾 → ([(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
148, 13sbceqbid 3705 . . . . . . . 8 (𝑘 = 𝐾 → ([((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
1514sbcbidv 3753 . . . . . . 7 (𝑘 = 𝐾 → ([(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
1615sbcbidv 3753 . . . . . 6 (𝑘 = 𝐾 → ([(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
176, 16sbceqbid 3705 . . . . 5 (𝑘 = 𝐾 → ([((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) ↔ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
1817abbidv 2822 . . . 4 (𝑘 = 𝐾 → {𝑎[((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))} = {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))})
194, 18mpteq12dv 5120 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎[((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))
20 df-hdmap1 39395 . . 3 HDMap1 = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎[((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))
2119, 20, 3mptfvmpt 6987 . 2 (𝐾 ∈ V → (HDMap1‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))
221, 21syl 17 1 (𝐾𝑋 → (HDMap1‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409  [wsbc 3698  ifcif 4423  {csn 4525   ↦ cmpt 5115   × cxp 5525  ‘cfv 6339  ℩crio 7112  (class class class)co 7155  1st c1st 7696  2nd c2nd 7697  Basecbs 16546  0gc0g 16776  -gcsg 18176  LSpanclspn 19816  LHypclh 37586  DVecHcdvh 38680  LCDualclcd 39188  mapdcmpd 39226  HDMap1chdma1 39393 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-hdmap1 39395 This theorem is referenced by:  hdmap1fval  39398
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