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Theorem hdmap1ffval 41192
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 = (LHypβ€˜πΎ)
Assertion
Ref Expression
hdmap1ffval (𝐾 ∈ 𝑋 β†’ (HDMap1β€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))}))
Distinct variable groups:   𝑀,𝐻   π‘Ž,𝑐,𝑑,𝑗,π‘š,𝑛,𝑒,𝑣,𝑀,𝐾   β„Ž,π‘Ž,π‘₯,𝑐,𝑑,𝑗,π‘š,𝑛,𝑒,𝑣,𝑀
Allowed substitution hints:   𝐻(π‘₯,𝑣,𝑒,β„Ž,𝑗,π‘š,𝑛,π‘Ž,𝑐,𝑑)   𝐾(π‘₯,β„Ž)   𝑋(π‘₯,𝑀,𝑣,𝑒,β„Ž,𝑗,π‘š,𝑛,π‘Ž,𝑐,𝑑)

Proof of Theorem hdmap1ffval
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3488 . 2 (𝐾 ∈ 𝑋 β†’ 𝐾 ∈ V)
2 fveq2 6891 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 hdmap1val.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2785 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6891 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DVecHβ€˜π‘˜) = (DVecHβ€˜πΎ))
65fveq1d 6893 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DVecHβ€˜π‘˜)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘€))
7 fveq2 6891 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (LCDualβ€˜π‘˜) = (LCDualβ€˜πΎ))
87fveq1d 6893 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((LCDualβ€˜π‘˜)β€˜π‘€) = ((LCDualβ€˜πΎ)β€˜π‘€))
9 fveq2 6891 . . . . . . . . . . . . 13 (π‘˜ = 𝐾 β†’ (mapdβ€˜π‘˜) = (mapdβ€˜πΎ))
109fveq1d 6893 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ ((mapdβ€˜π‘˜)β€˜π‘€) = ((mapdβ€˜πΎ)β€˜π‘€))
1110sbceq1d 3779 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ ([((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
1211sbcbidv 3833 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ ([(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
1312sbcbidv 3833 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ([(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
148, 13sbceqbid 3781 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ([((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
1514sbcbidv 3833 . . . . . . 7 (π‘˜ = 𝐾 β†’ ([(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
1615sbcbidv 3833 . . . . . 6 (π‘˜ = 𝐾 β†’ ([(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
176, 16sbceqbid 3781 . . . . 5 (π‘˜ = 𝐾 β†’ ([((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) ↔ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
1817abbidv 2796 . . . 4 (π‘˜ = 𝐾 β†’ {π‘Ž ∣ [((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))} = {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))})
194, 18mpteq12dv 5233 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {π‘Ž ∣ [((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))}) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))}))
20 df-hdmap1 41190 . . 3 HDMap1 = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {π‘Ž ∣ [((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜π‘˜)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))}))
2119, 20, 3mptfvmpt 7234 . 2 (𝐾 ∈ V β†’ (HDMap1β€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))}))
221, 21syl 17 1 (𝐾 ∈ 𝑋 β†’ (HDMap1β€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘€) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2704  Vcvv 3469  [wsbc 3774  ifcif 4524  {csn 4624   ↦ cmpt 5225   Γ— cxp 5670  β€˜cfv 6542  β„©crio 7369  (class class class)co 7414  1st c1st 7983  2nd c2nd 7984  Basecbs 17165  0gc0g 17406  -gcsg 18877  LSpanclspn 20837  LHypclh 39381  DVecHcdvh 40475  LCDualclcd 40983  mapdcmpd 41021  HDMap1chdma1 41188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-hdmap1 41190
This theorem is referenced by:  hdmap1fval  41193
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