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Theorem hdmap1fval 37578
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 37577 . . . . 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 6413 . . . 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 2859 . . 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 6411 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 fveq2 6411 . . . . . . . . . 10 (𝑤 = 𝑊 → ((LCDual‘𝐾)‘𝑤) = ((LCDual‘𝐾)‘𝑊))
9 fveq2 6411 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((mapd‘𝐾)‘𝑤) = ((mapd‘𝐾)‘𝑊))
109sbceq1d 3645 . . . . . . . . . . . 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 3695 . . . . . . . . . . 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 3695 . . . . . . . . . 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 3647 . . . . . . . . 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 3695 . . . . . . . 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 3695 . . . . . . 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 3647 . . . . . 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 6424 . . . . . . 7 ((DVecH‘𝐾)‘𝑊) ∈ V
18 fvex 6424 . . . . . . 7 (Base‘𝑢) ∈ V
19 fvex 6424 . . . . . . 7 (LSpan‘𝑢) ∈ V
20 hdmap1fval.u . . . . . . . . . . 11 𝑈 = ((DVecH‘𝐾)‘𝑊)
2120eqeq2i 2825 . . . . . . . . . 10 (𝑢 = 𝑈𝑢 = ((DVecH‘𝐾)‘𝑊))
2221biimpri 219 . . . . . . . . 9 (𝑢 = ((DVecH‘𝐾)‘𝑊) → 𝑢 = 𝑈)
23223ad2ant1 1156 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑢 = 𝑈)
24 simp2 1160 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = (Base‘𝑢))
2522fveq2d 6415 . . . . . . . . . . 11 (𝑢 = ((DVecH‘𝐾)‘𝑊) → (Base‘𝑢) = (Base‘𝑈))
26253ad2ant1 1156 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → (Base‘𝑢) = (Base‘𝑈))
2724, 26eqtrd 2847 . . . . . . . . 9 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = (Base‘𝑈))
28 hdmap1fval.v . . . . . . . . 9 𝑉 = (Base‘𝑈)
2927, 28syl6eqr 2865 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = 𝑉)
30 simp3 1161 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = (LSpan‘𝑢))
3123fveq2d 6415 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → (LSpan‘𝑢) = (LSpan‘𝑈))
3230, 31eqtrd 2847 . . . . . . . . 9 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = (LSpan‘𝑈))
33 hdmap1fval.n . . . . . . . . 9 𝑁 = (LSpan‘𝑈)
3432, 33syl6eqr 2865 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = 𝑁)
35 fvex 6424 . . . . . . . . . 10 ((LCDual‘𝐾)‘𝑊) ∈ V
36 fvex 6424 . . . . . . . . . 10 (Base‘𝑐) ∈ V
37 fvex 6424 . . . . . . . . . 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 2865 . . . . . . . . . . . 12 (𝑐 = ((LCDual‘𝐾)‘𝑊) → 𝑐 = 𝐶)
41403ad2ant1 1156 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑐 = 𝐶)
42 simp2 1160 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑑 = (Base‘𝑐))
4341fveq2d 6415 . . . . . . . . . . . . 13 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (Base‘𝑐) = (Base‘𝐶))
44 hdmap1fval.d . . . . . . . . . . . . 13 𝐷 = (Base‘𝐶)
4543, 44syl6eqr 2865 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (Base‘𝑐) = 𝐷)
4642, 45eqtrd 2847 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑑 = 𝐷)
47 simp3 1161 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑗 = (LSpan‘𝑐))
4841fveq2d 6415 . . . . . . . . . . . . 13 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (LSpan‘𝑐) = (LSpan‘𝐶))
49 hdmap1fval.j . . . . . . . . . . . . 13 𝐽 = (LSpan‘𝐶)
5048, 49syl6eqr 2865 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (LSpan‘𝑐) = 𝐽)
5147, 50eqtrd 2847 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑗 = 𝐽)
52 fvex 6424 . . . . . . . . . . . . 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 2865 . . . . . . . . . . . . . 14 (𝑚 = ((mapd‘𝐾)‘𝑊) → 𝑚 = 𝑀)
56 fveq1 6410 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑀 → (𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑀‘(𝑛‘{(2nd𝑥)})))
5756eqeq1d 2815 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ↔ (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{})))
58 fveq1 6410 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑀 → (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})))
5958eqeq1d 2815 . . . . . . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . . . . 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 4305 . . . . . . . . . . . . . . . 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 4946 . . . . . . . . . . . . . . 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 2878 . . . . . . . . . . . . . 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 3675 . . . . . . . . . . . 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 1160 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑑 = 𝐷)
68 xpeq2 5338 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → (𝑣 × 𝑑) = (𝑣 × 𝐷))
6968xpeq1d 5346 . . . . . . . . . . . . . . 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 1159 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑐 = 𝐶)
7271fveq2d 6415 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (0g𝑐) = (0g𝐶))
73 hdmap1fval.q . . . . . . . . . . . . . . . 16 𝑄 = (0g𝐶)
7472, 73syl6eqr 2865 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (0g𝑐) = 𝑄)
75 simp3 1161 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑗 = 𝐽)
7675fveq1d 6413 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑗‘{}) = (𝐽‘{}))
7776eqeq2d 2823 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ↔ (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{})))
7871fveq2d 6415 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (-g𝑐) = (-g𝐶))
79 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (-g𝐶)
8078, 79syl6eqr 2865 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (-g𝑐) = 𝑅)
8180oveqd 6894 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((2nd ‘(1st𝑥))(-g𝑐)) = ((2nd ‘(1st𝑥))𝑅))
8281sneqd 4389 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → {((2nd ‘(1st𝑥))(-g𝑐))} = {((2nd ‘(1st𝑥))𝑅)})
8375, 82fveq12d 6418 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))
8483eqeq2d 2823 . . . . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . . . . 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 6841 . . . . . . . . . . . . . . 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 4306 . . . . . . . . . . . . . 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 4934 . . . . . . . . . . . . 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 2878 . . . . . . . . . . . 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 274 . . . . . . . . . . 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 1483 . . . . . . . . . 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 3710 . . . . . . . . 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 1160 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑣 = 𝑉)
9493xpeq1d 5346 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑣 × 𝐷) = (𝑉 × 𝐷))
9594, 93xpeq12d 5348 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑣 × 𝐷) × 𝑣) = ((𝑉 × 𝐷) × 𝑉))
96 simp1 1159 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑢 = 𝑈)
9796fveq2d 6415 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (0g𝑢) = (0g𝑈))
98 hdmap1fval.o . . . . . . . . . . . . . 14 0 = (0g𝑈)
9997, 98syl6eqr 2865 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (0g𝑢) = 0 )
10099eqeq2d 2823 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((2nd𝑥) = (0g𝑢) ↔ (2nd𝑥) = 0 ))
101 simp3 1161 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑛 = 𝑁)
102101fveq1d 6413 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑛‘{(2nd𝑥)}) = (𝑁‘{(2nd𝑥)}))
103102fveqeq2d 6419 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{})))
10496fveq2d 6415 . . . . . . . . . . . . . . . . . . 19 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (-g𝑢) = (-g𝑈))
105 hdmap1fval.s . . . . . . . . . . . . . . . . . . 19 = (-g𝑈)
106104, 105syl6eqr 2865 . . . . . . . . . . . . . . . . . 18 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (-g𝑢) = )
107106oveqd 6894 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥)) = ((1st ‘(1st𝑥)) (2nd𝑥)))
108107sneqd 4389 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → {((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))} = {((1st ‘(1st𝑥)) (2nd𝑥))})
109101, 108fveq12d 6418 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))}) = (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}))
110109fveqeq2d 6419 . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . 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 6840 . . . . . . . . . . . 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 4311 . . . . . . . . . . 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 4934 . . . . . . . . . 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 2878 . . . . . . . . 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 274 . . . . . . . 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 1483 . . . . . . 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 3710 . . . . . 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 278 . . . . 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 2934 . . . 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 2813 . . . 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 6425 . . . . . . 7 𝑉 ∈ V
12344fvexi 6425 . . . . . . 7 𝐷 ∈ V
124122, 123xpex 7195 . . . . . 6 (𝑉 × 𝐷) ∈ V
125124, 122xpex 7195 . . . . 5 ((𝑉 × 𝐷) × 𝑉) ∈ V
126125mptex 6714 . . . 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 6506 . . 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 2867 . 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  {cab 2799  [wsbc 3640  ifcif 4286  {csn 4377  cmpt 4930   × cxp 5316  cfv 6104  crio 6837  (class class class)co 6877  1st c1st 7399  2nd c2nd 7400  Basecbs 16071  0gc0g 16308  -gcsg 17632  LSpanclspn 19181  LHypclh 35766  DVecHcdvh 36860  LCDualclcd 37368  mapdcmpd 37406  HDMap1chdma1 37573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-hdmap1 37575
This theorem is referenced by:  hdmap1vallem  37579
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