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Theorem hdmap1fval 38355
 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 = (LHyp‘𝐾)
hdmap1fval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1fval.v 𝑉 = (Base‘𝑈)
hdmap1fval.s = (-g𝑈)
hdmap1fval.o 0 = (0g𝑈)
hdmap1fval.n 𝑁 = (LSpan‘𝑈)
hdmap1fval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1fval.d 𝐷 = (Base‘𝐶)
hdmap1fval.r 𝑅 = (-g𝐶)
hdmap1fval.q 𝑄 = (0g𝐶)
hdmap1fval.j 𝐽 = (LSpan‘𝐶)
hdmap1fval.m 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1fval.i 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1fval.k (𝜑 → (𝐾𝐴𝑊𝐻))
Assertion
Ref Expression
hdmap1fval (𝜑𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
Distinct variable groups:   𝑥,,𝐶   𝐷,,𝑥   ,𝐽,𝑥   ,𝑀,𝑥   ,𝑁,𝑥   𝑈,,𝑥   ,𝑉,𝑥
Allowed substitution hints:   𝜑(𝑥,)   𝐴(𝑥,)   𝑄(𝑥,)   𝑅(𝑥,)   𝐻(𝑥,)   𝐼(𝑥,)   𝐾(𝑥,)   (𝑥,)   𝑊(𝑥,)   0 (𝑥,)

Proof of Theorem hdmap1fval
Dummy variables 𝑤 𝑎 𝑐 𝑑 𝑗 𝑚 𝑛 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1fval.k . 2 (𝜑 → (𝐾𝐴𝑊𝐻))
2 hdmap1fval.i . . . 4 𝐼 = ((HDMap1‘𝐾)‘𝑊)
3 hdmap1val.h . . . . . 6 𝐻 = (LHyp‘𝐾)
43hdmap1ffval 38354 . . . . 5 (𝐾𝐴 → (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))})))))}))
54fveq1d 6499 . . . 4 (𝐾𝐴 → ((HDMap1‘𝐾)‘𝑊) = ((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))})))))})‘𝑊))
62, 5syl5eq 2823 . . 3 (𝐾𝐴𝐼 = ((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))})))))})‘𝑊))
7 fveq2 6497 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 fveq2 6497 . . . . . . . . . 10 (𝑤 = 𝑊 → ((LCDual‘𝐾)‘𝑤) = ((LCDual‘𝐾)‘𝑊))
9 fveq2 6497 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((mapd‘𝐾)‘𝑤) = ((mapd‘𝐾)‘𝑊))
109sbceq1d 3685 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ([((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))})))))))
1110sbcbidv 3730 . . . . . . . . . . 11 (𝑤 = 𝑊 → ([(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))})))))))
1211sbcbidv 3730 . . . . . . . . . 10 (𝑤 = 𝑊 → ([(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))})))))))
138, 12sbceqbid 3687 . . . . . . . . 9 (𝑤 = 𝑊 → ([((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))})))))))
1413sbcbidv 3730 . . . . . . . 8 (𝑤 = 𝑊 → ([(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))})))))))
1514sbcbidv 3730 . . . . . . 7 (𝑤 = 𝑊 → ([(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))})))))))
167, 15sbceqbid 3687 . . . . . 6 (𝑤 = 𝑊 → ([((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))})))))))
17 fvex 6510 . . . . . . 7 ((DVecH‘𝐾)‘𝑊) ∈ V
18 fvex 6510 . . . . . . 7 (Base‘𝑢) ∈ V
19 fvex 6510 . . . . . . 7 (LSpan‘𝑢) ∈ V
20 hdmap1fval.u . . . . . . . . . . 11 𝑈 = ((DVecH‘𝐾)‘𝑊)
2120eqeq2i 2787 . . . . . . . . . 10 (𝑢 = 𝑈𝑢 = ((DVecH‘𝐾)‘𝑊))
2221biimpri 220 . . . . . . . . 9 (𝑢 = ((DVecH‘𝐾)‘𝑊) → 𝑢 = 𝑈)
23223ad2ant1 1113 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑢 = 𝑈)
24 simp2 1117 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = (Base‘𝑢))
2522fveq2d 6501 . . . . . . . . . . 11 (𝑢 = ((DVecH‘𝐾)‘𝑊) → (Base‘𝑢) = (Base‘𝑈))
26253ad2ant1 1113 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → (Base‘𝑢) = (Base‘𝑈))
2724, 26eqtrd 2811 . . . . . . . . 9 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = (Base‘𝑈))
28 hdmap1fval.v . . . . . . . . 9 𝑉 = (Base‘𝑈)
2927, 28syl6eqr 2829 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = 𝑉)
30 simp3 1118 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = (LSpan‘𝑢))
3123fveq2d 6501 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → (LSpan‘𝑢) = (LSpan‘𝑈))
3230, 31eqtrd 2811 . . . . . . . . 9 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = (LSpan‘𝑈))
33 hdmap1fval.n . . . . . . . . 9 𝑁 = (LSpan‘𝑈)
3432, 33syl6eqr 2829 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = 𝑁)
35 fvex 6510 . . . . . . . . . 10 ((LCDual‘𝐾)‘𝑊) ∈ V
36 fvex 6510 . . . . . . . . . 10 (Base‘𝑐) ∈ V
37 fvex 6510 . . . . . . . . . 10 (LSpan‘𝑐) ∈ V
38 id 22 . . . . . . . . . . . . 13 (𝑐 = ((LCDual‘𝐾)‘𝑊) → 𝑐 = ((LCDual‘𝐾)‘𝑊))
39 hdmap1fval.c . . . . . . . . . . . . 13 𝐶 = ((LCDual‘𝐾)‘𝑊)
4038, 39syl6eqr 2829 . . . . . . . . . . . 12 (𝑐 = ((LCDual‘𝐾)‘𝑊) → 𝑐 = 𝐶)
41403ad2ant1 1113 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑐 = 𝐶)
42 simp2 1117 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑑 = (Base‘𝑐))
4341fveq2d 6501 . . . . . . . . . . . . 13 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (Base‘𝑐) = (Base‘𝐶))
44 hdmap1fval.d . . . . . . . . . . . . 13 𝐷 = (Base‘𝐶)
4543, 44syl6eqr 2829 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (Base‘𝑐) = 𝐷)
4642, 45eqtrd 2811 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑑 = 𝐷)
47 simp3 1118 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑗 = (LSpan‘𝑐))
4841fveq2d 6501 . . . . . . . . . . . . 13 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (LSpan‘𝑐) = (LSpan‘𝐶))
49 hdmap1fval.j . . . . . . . . . . . . 13 𝐽 = (LSpan‘𝐶)
5048, 49syl6eqr 2829 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (LSpan‘𝑐) = 𝐽)
5147, 50eqtrd 2811 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑗 = 𝐽)
52 fvex 6510 . . . . . . . . . . . . 13 ((mapd‘𝐾)‘𝑊) ∈ V
53 id 22 . . . . . . . . . . . . . . 15 (𝑚 = ((mapd‘𝐾)‘𝑊) → 𝑚 = ((mapd‘𝐾)‘𝑊))
54 hdmap1fval.m . . . . . . . . . . . . . . 15 𝑀 = ((mapd‘𝐾)‘𝑊)
5553, 54syl6eqr 2829 . . . . . . . . . . . . . 14 (𝑚 = ((mapd‘𝐾)‘𝑊) → 𝑚 = 𝑀)
56 fveq1 6496 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑀 → (𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑀‘(𝑛‘{(2nd𝑥)})))
5756eqeq1d 2777 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ↔ (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{})))
58 fveq1 6496 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑀 → (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})))
5958eqeq1d 2777 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → ((𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}) ↔ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))
6057, 59anbi12d 621 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑀 → (((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})) ↔ ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))
6160riotabidv 6937 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑀 → (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))) = (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))
6261ifeq2d 4367 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀 → if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))) = if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))
6362mpteq2dv 5021 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) = (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))))
6463eleq2d 2848 . . . . . . . . . . . . . 14 (𝑚 = 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 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
6555, 64syl 17 . . . . . . . . . . . . 13 (𝑚 = ((mapd‘𝐾)‘𝑊) → (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 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))))
6652, 65sbcie 3715 . . . . . . . . . . . 12 ([((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 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))))
67 simp2 1117 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑑 = 𝐷)
68 xpeq2 5425 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → (𝑣 × 𝑑) = (𝑣 × 𝐷))
6968xpeq1d 5433 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → ((𝑣 × 𝑑) × 𝑣) = ((𝑣 × 𝐷) × 𝑣))
7067, 69syl 17 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((𝑣 × 𝑑) × 𝑣) = ((𝑣 × 𝐷) × 𝑣))
71 simp1 1116 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑐 = 𝐶)
7271fveq2d 6501 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (0g𝑐) = (0g𝐶))
73 hdmap1fval.q . . . . . . . . . . . . . . . 16 𝑄 = (0g𝐶)
7472, 73syl6eqr 2829 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (0g𝑐) = 𝑄)
75 simp3 1118 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑗 = 𝐽)
7675fveq1d 6499 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑗‘{}) = (𝐽‘{}))
7776eqeq2d 2785 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ↔ (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{})))
7871fveq2d 6501 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (-g𝑐) = (-g𝐶))
79 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (-g𝐶)
8078, 79syl6eqr 2829 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (-g𝑐) = 𝑅)
8180oveqd 6991 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((2nd ‘(1st𝑥))(-g𝑐)) = ((2nd ‘(1st𝑥))𝑅))
8281sneqd 4451 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → {((2nd ‘(1st𝑥))(-g𝑐))} = {((2nd ‘(1st𝑥))𝑅)})
8375, 82fveq12d 6504 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))
8483eqeq2d 2785 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}) ↔ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))
8577, 84anbi12d 621 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})) ↔ ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))
8667, 85riotaeqbidv 6938 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))) = (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))
8774, 86ifeq12d 4368 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))) = if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
8870, 87mpteq12dv 5010 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}))))) = (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
8988eleq2d 2848 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (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 ∈ (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
9066, 89syl5bb 275 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 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 ∈ (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
9141, 46, 51, 90syl3anc 1351 . . . . . . . . . 10 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → ([((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 ∈ (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
9235, 36, 37, 91sbc3ie 3754 . . . . . . . . 9 ([((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 ∈ (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
93 simp2 1117 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑣 = 𝑉)
9493xpeq1d 5433 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑣 × 𝐷) = (𝑉 × 𝐷))
9594, 93xpeq12d 5435 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑣 × 𝐷) × 𝑣) = ((𝑉 × 𝐷) × 𝑉))
96 simp1 1116 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑢 = 𝑈)
9796fveq2d 6501 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (0g𝑢) = (0g𝑈))
98 hdmap1fval.o . . . . . . . . . . . . . 14 0 = (0g𝑈)
9997, 98syl6eqr 2829 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (0g𝑢) = 0 )
10099eqeq2d 2785 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((2nd𝑥) = (0g𝑢) ↔ (2nd𝑥) = 0 ))
101 simp3 1118 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑛 = 𝑁)
102101fveq1d 6499 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑛‘{(2nd𝑥)}) = (𝑁‘{(2nd𝑥)}))
103102fveqeq2d 6505 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{})))
10496fveq2d 6501 . . . . . . . . . . . . . . . . . . 19 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (-g𝑢) = (-g𝑈))
105 hdmap1fval.s . . . . . . . . . . . . . . . . . . 19 = (-g𝑈)
106104, 105syl6eqr 2829 . . . . . . . . . . . . . . . . . 18 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (-g𝑢) = )
107106oveqd 6991 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥)) = ((1st ‘(1st𝑥)) (2nd𝑥)))
108107sneqd 4451 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → {((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))} = {((1st ‘(1st𝑥)) (2nd𝑥))})
109101, 108fveq12d 6504 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))}) = (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}))
110109fveqeq2d 6505 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))
111103, 110anbi12d 621 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})) ↔ ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))
112111riotabidv 6937 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))
113100, 112ifbieq2d 4373 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))) = if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
11495, 113mpteq12dv 5010 . . . . . . . . . 10 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))) = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
115114eleq2d 2848 . . . . . . . . 9 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝐷) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), 𝑄, (𝐷 ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))) ↔ 𝑎 ∈ (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
11692, 115syl5bb 275 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 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 ∈ (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
11723, 29, 34, 116syl3anc 1351 . . . . . . 7 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → ([((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 ∈ (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
11817, 18, 19, 117sbc3ie 3754 . . . . . 6 ([((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 ∈ (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
11916, 118syl6bb 279 . . . . 5 (𝑤 = 𝑊 → ([((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 ∈ (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))))
120119abbi1dv 2901 . . . 4 (𝑤 = 𝑊 → {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))})))))} = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
121 eqid 2775 . . . 4 (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))})))))}) = (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))})))))})
12228fvexi 6511 . . . . . . 7 𝑉 ∈ V
12344fvexi 6511 . . . . . . 7 𝐷 ∈ V
124122, 123xpex 7291 . . . . . 6 (𝑉 × 𝐷) ∈ V
125124, 122xpex 7291 . . . . 5 ((𝑉 × 𝐷) × 𝑉) ∈ V
126125mptex 6810 . . . 4 (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))) ∈ V
127120, 121, 126fvmpt 6593 . . 3 (𝑊𝐻 → ((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))})))))})‘𝑊) = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
1286, 127sylan9eq 2831 . 2 ((𝐾𝐴𝑊𝐻) → 𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
1291, 128syl 17 1 (𝜑𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048  {cab 2755  [wsbc 3680  ifcif 4348  {csn 4439   ↦ cmpt 5006   × cxp 5402  ‘cfv 6186  ℩crio 6934  (class class class)co 6974  1st c1st 7496  2nd c2nd 7497  Basecbs 16333  0gc0g 16563  -gcsg 17887  LSpanclspn 19459  LHypclh 36543  DVecHcdvh 37637  LCDualclcd 38145  mapdcmpd 38183  HDMap1chdma1 38350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-rep 5047  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-ral 3090  df-rex 3091  df-reu 3092  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6935  df-ov 6977  df-hdmap1 38352 This theorem is referenced by:  hdmap1vallem  38356
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