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Theorem hdmap1fval 41324
Description: Preliminary map from vectors to functionals in the closed kernel dual space. TODO: change span 𝐽 to the convention 𝐿 for this section. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1val.h 𝐻 = (LHypβ€˜πΎ)
hdmap1fval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmap1fval.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmap1fval.s βˆ’ = (-gβ€˜π‘ˆ)
hdmap1fval.o 0 = (0gβ€˜π‘ˆ)
hdmap1fval.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmap1fval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmap1fval.d 𝐷 = (Baseβ€˜πΆ)
hdmap1fval.r 𝑅 = (-gβ€˜πΆ)
hdmap1fval.q 𝑄 = (0gβ€˜πΆ)
hdmap1fval.j 𝐽 = (LSpanβ€˜πΆ)
hdmap1fval.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
hdmap1fval.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmap1fval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
hdmap1fval (πœ‘ β†’ 𝐼 = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
Distinct variable groups:   π‘₯,β„Ž,𝐢   𝐷,β„Ž,π‘₯   β„Ž,𝐽,π‘₯   β„Ž,𝑀,π‘₯   β„Ž,𝑁,π‘₯   π‘ˆ,β„Ž,π‘₯   β„Ž,𝑉,π‘₯
Allowed substitution hints:   πœ‘(π‘₯,β„Ž)   𝐴(π‘₯,β„Ž)   𝑄(π‘₯,β„Ž)   𝑅(π‘₯,β„Ž)   𝐻(π‘₯,β„Ž)   𝐼(π‘₯,β„Ž)   𝐾(π‘₯,β„Ž)   βˆ’ (π‘₯,β„Ž)   π‘Š(π‘₯,β„Ž)   0 (π‘₯,β„Ž)

Proof of Theorem hdmap1fval
Dummy variables 𝑀 π‘Ž 𝑐 𝑑 𝑗 π‘š 𝑛 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1fval.k . 2 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
2 hdmap1fval.i . . . 4 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
3 hdmap1val.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
43hdmap1ffval 41323 . . . . 5 (𝐾 ∈ 𝐴 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)})))))}))
54fveq1d 6893 . . . 4 (𝐾 ∈ 𝐴 β†’ ((HDMap1β€˜πΎ)β€˜π‘Š) = ((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)})))))})β€˜π‘Š))
62, 5eqtrid 2777 . . 3 (𝐾 ∈ 𝐴 β†’ 𝐼 = ((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)})))))})β€˜π‘Š))
7 fveq2 6891 . . . . . . 7 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘Š))
8 fveq2 6891 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ((LCDualβ€˜πΎ)β€˜π‘€) = ((LCDualβ€˜πΎ)β€˜π‘Š))
9 fveq2 6891 . . . . . . . . . . . . 13 (𝑀 = π‘Š β†’ ((mapdβ€˜πΎ)β€˜π‘€) = ((mapdβ€˜πΎ)β€˜π‘Š))
109sbceq1d 3774 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ ([((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)}))))) ↔ [((mapdβ€˜πΎ)β€˜π‘Š) / 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)})))))))
1110sbcbidv 3829 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ([(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)}))))) ↔ [(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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 3829 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ([(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)}))))) ↔ [(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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)})))))))
138, 12sbceqbid 3776 . . . . . . . . 9 (𝑀 = π‘Š β†’ ([((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)}))))) ↔ [((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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)})))))))
1413sbcbidv 3829 . . . . . . . 8 (𝑀 = π‘Š β†’ ([(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)}))))) ↔ [(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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 3829 . . . . . . 7 (𝑀 = π‘Š β†’ ([(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)}))))) ↔ [(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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)})))))))
167, 15sbceqbid 3776 . . . . . 6 (𝑀 = π‘Š β†’ ([((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)}))))) ↔ [((DVecHβ€˜πΎ)β€˜π‘Š) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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)})))))))
17 fvex 6904 . . . . . . 7 ((DVecHβ€˜πΎ)β€˜π‘Š) ∈ V
18 fvex 6904 . . . . . . 7 (Baseβ€˜π‘’) ∈ V
19 fvex 6904 . . . . . . 7 (LSpanβ€˜π‘’) ∈ V
20 hdmap1fval.u . . . . . . . . . . 11 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
2120eqeq2i 2738 . . . . . . . . . 10 (𝑒 = π‘ˆ ↔ 𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š))
2221biimpri 227 . . . . . . . . 9 (𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) β†’ 𝑒 = π‘ˆ)
23223ad2ant1 1130 . . . . . . . 8 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑒 = π‘ˆ)
24 simp2 1134 . . . . . . . . . 10 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑣 = (Baseβ€˜π‘’))
2522fveq2d 6895 . . . . . . . . . . 11 (𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) β†’ (Baseβ€˜π‘’) = (Baseβ€˜π‘ˆ))
26253ad2ant1 1130 . . . . . . . . . 10 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ (Baseβ€˜π‘’) = (Baseβ€˜π‘ˆ))
2724, 26eqtrd 2765 . . . . . . . . 9 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑣 = (Baseβ€˜π‘ˆ))
28 hdmap1fval.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘ˆ)
2927, 28eqtr4di 2783 . . . . . . . 8 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑣 = 𝑉)
30 simp3 1135 . . . . . . . . . 10 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑛 = (LSpanβ€˜π‘’))
3123fveq2d 6895 . . . . . . . . . 10 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ (LSpanβ€˜π‘’) = (LSpanβ€˜π‘ˆ))
3230, 31eqtrd 2765 . . . . . . . . 9 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑛 = (LSpanβ€˜π‘ˆ))
33 hdmap1fval.n . . . . . . . . 9 𝑁 = (LSpanβ€˜π‘ˆ)
3432, 33eqtr4di 2783 . . . . . . . 8 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ 𝑛 = 𝑁)
35 fvex 6904 . . . . . . . . . 10 ((LCDualβ€˜πΎ)β€˜π‘Š) ∈ V
36 fvex 6904 . . . . . . . . . 10 (Baseβ€˜π‘) ∈ V
37 fvex 6904 . . . . . . . . . 10 (LSpanβ€˜π‘) ∈ V
38 id 22 . . . . . . . . . . . . 13 (𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) β†’ 𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š))
39 hdmap1fval.c . . . . . . . . . . . . 13 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
4038, 39eqtr4di 2783 . . . . . . . . . . . 12 (𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) β†’ 𝑐 = 𝐢)
41403ad2ant1 1130 . . . . . . . . . . 11 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ 𝑐 = 𝐢)
42 simp2 1134 . . . . . . . . . . . 12 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ 𝑑 = (Baseβ€˜π‘))
4341fveq2d 6895 . . . . . . . . . . . . 13 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
44 hdmap1fval.d . . . . . . . . . . . . 13 𝐷 = (Baseβ€˜πΆ)
4543, 44eqtr4di 2783 . . . . . . . . . . . 12 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ (Baseβ€˜π‘) = 𝐷)
4642, 45eqtrd 2765 . . . . . . . . . . 11 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ 𝑑 = 𝐷)
47 simp3 1135 . . . . . . . . . . . 12 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ 𝑗 = (LSpanβ€˜π‘))
4841fveq2d 6895 . . . . . . . . . . . . 13 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ (LSpanβ€˜π‘) = (LSpanβ€˜πΆ))
49 hdmap1fval.j . . . . . . . . . . . . 13 𝐽 = (LSpanβ€˜πΆ)
5048, 49eqtr4di 2783 . . . . . . . . . . . 12 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ (LSpanβ€˜π‘) = 𝐽)
5147, 50eqtrd 2765 . . . . . . . . . . 11 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ 𝑗 = 𝐽)
52 fvex 6904 . . . . . . . . . . . . 13 ((mapdβ€˜πΎ)β€˜π‘Š) ∈ V
53 id 22 . . . . . . . . . . . . . . 15 (π‘š = ((mapdβ€˜πΎ)β€˜π‘Š) β†’ π‘š = ((mapdβ€˜πΎ)β€˜π‘Š))
54 hdmap1fval.m . . . . . . . . . . . . . . 15 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
5553, 54eqtr4di 2783 . . . . . . . . . . . . . 14 (π‘š = ((mapdβ€˜πΎ)β€˜π‘Š) β†’ π‘š = 𝑀)
56 fveq1 6890 . . . . . . . . . . . . . . . . . . . 20 (π‘š = 𝑀 β†’ (π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})))
5756eqeq1d 2727 . . . . . . . . . . . . . . . . . . 19 (π‘š = 𝑀 β†’ ((π‘šβ€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ↔ (π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž})))
58 fveq1 6890 . . . . . . . . . . . . . . . . . . . 20 (π‘š = 𝑀 β†’ (π‘šβ€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})))
5958eqeq1d 2727 . . . . . . . . . . . . . . . . . . 19 (π‘š = 𝑀 β†’ ((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‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))
6057, 59anbi12d 630 . . . . . . . . . . . . . . . . . 18 (π‘š = 𝑀 β†’ (((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‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))
6160riotabidv 7373 . . . . . . . . . . . . . . . . 17 (π‘š = 𝑀 β†’ (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)}))) = (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))
6261ifeq2d 4544 . . . . . . . . . . . . . . . 16 (π‘š = 𝑀 β†’ 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)})))) = if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))
6362mpteq2dv 5245 . . . . . . . . . . . . . . 15 (π‘š = 𝑀 β†’ (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‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))))
6463eleq2d 2811 . . . . . . . . . . . . . 14 (π‘š = 𝑀 β†’ (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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
6555, 64syl 17 . . . . . . . . . . . . 13 (π‘š = ((mapdβ€˜πΎ)β€˜π‘Š) β†’ (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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))))))
6652, 65sbcie 3813 . . . . . . . . . . . 12 ([((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))))
67 simp2 1134 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ 𝑑 = 𝐷)
68 xpeq2 5693 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 β†’ (𝑣 Γ— 𝑑) = (𝑣 Γ— 𝐷))
6968xpeq1d 5701 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 β†’ ((𝑣 Γ— 𝑑) Γ— 𝑣) = ((𝑣 Γ— 𝐷) Γ— 𝑣))
7067, 69syl 17 . . . . . . . . . . . . . 14 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ ((𝑣 Γ— 𝑑) Γ— 𝑣) = ((𝑣 Γ— 𝐷) Γ— 𝑣))
71 simp1 1133 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ 𝑐 = 𝐢)
7271fveq2d 6895 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (0gβ€˜π‘) = (0gβ€˜πΆ))
73 hdmap1fval.q . . . . . . . . . . . . . . . 16 𝑄 = (0gβ€˜πΆ)
7472, 73eqtr4di 2783 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (0gβ€˜π‘) = 𝑄)
75 simp3 1135 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ 𝑗 = 𝐽)
7675fveq1d 6893 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (π‘—β€˜{β„Ž}) = (π½β€˜{β„Ž}))
7776eqeq2d 2736 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ↔ (π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž})))
7871fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (-gβ€˜π‘) = (-gβ€˜πΆ))
79 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (-gβ€˜πΆ)
8078, 79eqtr4di 2783 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (-gβ€˜π‘) = 𝑅)
8180oveqd 7432 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ ((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž) = ((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž))
8281sneqd 4636 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ {((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)} = {((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})
8375, 82fveq12d 6898 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))
8483eqeq2d 2736 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ ((π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}) ↔ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))
8577, 84anbi12d 630 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})) ↔ ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))
8667, 85riotaeqbidv 7374 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))) = (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))
8774, 86ifeq12d 4545 . . . . . . . . . . . . . 14 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)})))) = if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))
8870, 87mpteq12dv 5234 . . . . . . . . . . . . 13 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π‘—β€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜π‘)β„Ž)}))))) = (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
8988eleq2d 2811 . . . . . . . . . . . 12 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ (π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝑑) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), (0gβ€˜π‘), (β„©β„Ž ∈ 𝑑 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π‘—β€˜{β„Ž}) ∧ (π‘€β€˜(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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
9066, 89bitrid 282 . . . . . . . . . . 11 ((𝑐 = 𝐢 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽) β†’ ([((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
9141, 46, 51, 90syl3anc 1368 . . . . . . . . . 10 ((𝑐 = ((LCDualβ€˜πΎ)β€˜π‘Š) ∧ 𝑑 = (Baseβ€˜π‘) ∧ 𝑗 = (LSpanβ€˜π‘)) β†’ ([((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
9235, 36, 37, 91sbc3ie 3855 . . . . . . . . 9 ([((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
93 simp2 1134 . . . . . . . . . . . . 13 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ 𝑣 = 𝑉)
9493xpeq1d 5701 . . . . . . . . . . . 12 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (𝑣 Γ— 𝐷) = (𝑉 Γ— 𝐷))
9594, 93xpeq12d 5703 . . . . . . . . . . 11 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ ((𝑣 Γ— 𝐷) Γ— 𝑣) = ((𝑉 Γ— 𝐷) Γ— 𝑉))
96 simp1 1133 . . . . . . . . . . . . . . 15 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ 𝑒 = π‘ˆ)
9796fveq2d 6895 . . . . . . . . . . . . . 14 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (0gβ€˜π‘’) = (0gβ€˜π‘ˆ))
98 hdmap1fval.o . . . . . . . . . . . . . 14 0 = (0gβ€˜π‘ˆ)
9997, 98eqtr4di 2783 . . . . . . . . . . . . 13 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (0gβ€˜π‘’) = 0 )
10099eqeq2d 2736 . . . . . . . . . . . 12 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ ((2nd β€˜π‘₯) = (0gβ€˜π‘’) ↔ (2nd β€˜π‘₯) = 0 ))
101 simp3 1135 . . . . . . . . . . . . . . . 16 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ 𝑛 = 𝑁)
102101fveq1d 6893 . . . . . . . . . . . . . . 15 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (π‘›β€˜{(2nd β€˜π‘₯)}) = (π‘β€˜{(2nd β€˜π‘₯)}))
103102fveqeq2d 6899 . . . . . . . . . . . . . 14 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ↔ (π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž})))
10496fveq2d 6895 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (-gβ€˜π‘’) = (-gβ€˜π‘ˆ))
105 hdmap1fval.s . . . . . . . . . . . . . . . . . . 19 βˆ’ = (-gβ€˜π‘ˆ)
106104, 105eqtr4di 2783 . . . . . . . . . . . . . . . . . 18 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (-gβ€˜π‘’) = βˆ’ )
107106oveqd 7432 . . . . . . . . . . . . . . . . 17 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ ((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯)) = ((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯)))
108107sneqd 4636 . . . . . . . . . . . . . . . 16 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ {((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))} = {((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})
109101, 108fveq12d 6898 . . . . . . . . . . . . . . 15 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))}) = (π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))}))
110109fveqeq2d 6899 . . . . . . . . . . . . . 14 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ ((π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}) ↔ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))
111103, 110anbi12d 630 . . . . . . . . . . . . 13 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})) ↔ ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))
112111riotabidv 7373 . . . . . . . . . . . 12 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))) = (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))
113100, 112ifbieq2d 4550 . . . . . . . . . . 11 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))) = if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))
11495, 113mpteq12dv 5234 . . . . . . . . . 10 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))) = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
115114eleq2d 2811 . . . . . . . . 9 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ (π‘Ž ∈ (π‘₯ ∈ ((𝑣 Γ— 𝐷) Γ— 𝑣) ↦ if((2nd β€˜π‘₯) = (0gβ€˜π‘’), 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘›β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘›β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘’)(2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))) ↔ π‘Ž ∈ (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
11692, 115bitrid 282 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁) β†’ ([((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
11723, 29, 34, 116syl3anc 1368 . . . . . . 7 ((𝑒 = ((DVecHβ€˜πΎ)β€˜π‘Š) ∧ 𝑣 = (Baseβ€˜π‘’) ∧ 𝑛 = (LSpanβ€˜π‘’)) β†’ ([((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
11817, 18, 19, 117sbc3ie 3855 . . . . . 6 ([((DVecHβ€˜πΎ)β€˜π‘Š) / 𝑒][(Baseβ€˜π‘’) / 𝑣][(LSpanβ€˜π‘’) / 𝑛][((LCDualβ€˜πΎ)β€˜π‘Š) / 𝑐][(Baseβ€˜π‘) / 𝑑][(LSpanβ€˜π‘) / 𝑗][((mapdβ€˜πΎ)β€˜π‘Š) / 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 ∈ (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
11916, 118bitrdi 286 . . . . 5 (𝑀 = π‘Š β†’ ([((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)}))))) ↔ π‘Ž ∈ (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))))
120119eqabcdv 2860 . . . 4 (𝑀 = π‘Š β†’ {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)})))))} = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
121 eqid 2725 . . . 4 (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)})))))}) = (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)})))))})
12228fvexi 6905 . . . . . . 7 𝑉 ∈ V
12344fvexi 6905 . . . . . . 7 𝐷 ∈ V
124122, 123xpex 7752 . . . . . 6 (𝑉 Γ— 𝐷) ∈ V
125124, 122xpex 7752 . . . . 5 ((𝑉 Γ— 𝐷) Γ— 𝑉) ∈ V
126125mptex 7230 . . . 4 (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))) ∈ V
127120, 121, 126fvmpt 6999 . . 3 (π‘Š ∈ 𝐻 β†’ ((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)})))))})β€˜π‘Š) = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
1286, 127sylan9eq 2785 . 2 ((𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
1291, 128syl 17 1 (πœ‘ β†’ 𝐼 = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  [wsbc 3769  ifcif 4524  {csn 4624   ↦ cmpt 5226   Γ— cxp 5670  β€˜cfv 6542  β„©crio 7370  (class class class)co 7415  1st c1st 7987  2nd c2nd 7988  Basecbs 17177  0gc0g 17418  -gcsg 18894  LSpanclspn 20857  LHypclh 39512  DVecHcdvh 40606  LCDualclcd 41114  mapdcmpd 41152  HDMap1chdma1 41319
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  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-riota 7371  df-ov 7418  df-hdmap1 41321
This theorem is referenced by:  hdmap1vallem  41325
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