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Theorem hdmap1ffval 41778
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 3499 . 2 (𝐾𝑋𝐾 ∈ V)
2 fveq2 6907 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 hdmap1val.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2793 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6909 . . . . . 6 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
7 fveq2 6907 . . . . . . . . . 10 (𝑘 = 𝐾 → (LCDual‘𝑘) = (LCDual‘𝐾))
87fveq1d 6909 . . . . . . . . 9 (𝑘 = 𝐾 → ((LCDual‘𝑘)‘𝑤) = ((LCDual‘𝐾)‘𝑤))
9 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (mapd‘𝑘) = (mapd‘𝐾))
109fveq1d 6909 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((mapd‘𝑘)‘𝑤) = ((mapd‘𝐾)‘𝑤))
1110sbceq1d 3796 . . . . . . . . . . 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 3851 . . . . . . . . . 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 3851 . . . . . . . . 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 3798 . . . . . . . 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 3851 . . . . . . 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 3851 . . . . . 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 3798 . . . . 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 2806 . . . 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 5239 . . 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 41776 . . 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 7248 . 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 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  [wsbc 3791  ifcif 4531  {csn 4631  cmpt 5231   × cxp 5687  cfv 6563  crio 7387  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  0gc0g 17486  -gcsg 18966  LSpanclspn 20987  LHypclh 39967  DVecHcdvh 41061  LCDualclcd 41569  mapdcmpd 41607  HDMap1chdma1 41774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-hdmap1 41776
This theorem is referenced by:  hdmap1fval  41779
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