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Theorem hdmap1fval 36563
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 36562 . . . . 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 6150 . . . 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 2667 . . 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 6148 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 fveq2 6148 . . . . . . . . . 10 (𝑤 = 𝑊 → ((LCDual‘𝐾)‘𝑤) = ((LCDual‘𝐾)‘𝑊))
9 fveq2 6148 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((mapd‘𝐾)‘𝑤) = ((mapd‘𝐾)‘𝑊))
109sbceq1d 3422 . . . . . . . . . . . 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 3472 . . . . . . . . . . 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 3472 . . . . . . . . . 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 3424 . . . . . . . . 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 3472 . . . . . . . 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 3472 . . . . . . 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 3424 . . . . . 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 6158 . . . . . . 7 ((DVecH‘𝐾)‘𝑊) ∈ V
18 fvex 6158 . . . . . . 7 (Base‘𝑢) ∈ V
19 fvex 6158 . . . . . . 7 (LSpan‘𝑢) ∈ V
20 hdmap1fval.u . . . . . . . . . . 11 𝑈 = ((DVecH‘𝐾)‘𝑊)
2120eqeq2i 2633 . . . . . . . . . 10 (𝑢 = 𝑈𝑢 = ((DVecH‘𝐾)‘𝑊))
2221biimpri 218 . . . . . . . . 9 (𝑢 = ((DVecH‘𝐾)‘𝑊) → 𝑢 = 𝑈)
23223ad2ant1 1080 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑢 = 𝑈)
24 simp2 1060 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = (Base‘𝑢))
2522fveq2d 6152 . . . . . . . . . . 11 (𝑢 = ((DVecH‘𝐾)‘𝑊) → (Base‘𝑢) = (Base‘𝑈))
26253ad2ant1 1080 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → (Base‘𝑢) = (Base‘𝑈))
2724, 26eqtrd 2655 . . . . . . . . 9 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = (Base‘𝑈))
28 hdmap1fval.v . . . . . . . . 9 𝑉 = (Base‘𝑈)
2927, 28syl6eqr 2673 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑣 = 𝑉)
30 simp3 1061 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = (LSpan‘𝑢))
3123fveq2d 6152 . . . . . . . . . 10 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → (LSpan‘𝑢) = (LSpan‘𝑈))
3230, 31eqtrd 2655 . . . . . . . . 9 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = (LSpan‘𝑈))
33 hdmap1fval.n . . . . . . . . 9 𝑁 = (LSpan‘𝑈)
3432, 33syl6eqr 2673 . . . . . . . 8 ((𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢) ∧ 𝑛 = (LSpan‘𝑢)) → 𝑛 = 𝑁)
35 fvex 6158 . . . . . . . . . 10 ((LCDual‘𝐾)‘𝑊) ∈ V
36 fvex 6158 . . . . . . . . . 10 (Base‘𝑐) ∈ V
37 fvex 6158 . . . . . . . . . 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 2673 . . . . . . . . . . . 12 (𝑐 = ((LCDual‘𝐾)‘𝑊) → 𝑐 = 𝐶)
41403ad2ant1 1080 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑐 = 𝐶)
42 simp2 1060 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑑 = (Base‘𝑐))
4341fveq2d 6152 . . . . . . . . . . . . 13 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (Base‘𝑐) = (Base‘𝐶))
44 hdmap1fval.d . . . . . . . . . . . . 13 𝐷 = (Base‘𝐶)
4543, 44syl6eqr 2673 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (Base‘𝑐) = 𝐷)
4642, 45eqtrd 2655 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑑 = 𝐷)
47 simp3 1061 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑗 = (LSpan‘𝑐))
4841fveq2d 6152 . . . . . . . . . . . . 13 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (LSpan‘𝑐) = (LSpan‘𝐶))
49 hdmap1fval.j . . . . . . . . . . . . 13 𝐽 = (LSpan‘𝐶)
5048, 49syl6eqr 2673 . . . . . . . . . . . 12 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → (LSpan‘𝑐) = 𝐽)
5147, 50eqtrd 2655 . . . . . . . . . . 11 ((𝑐 = ((LCDual‘𝐾)‘𝑊) ∧ 𝑑 = (Base‘𝑐) ∧ 𝑗 = (LSpan‘𝑐)) → 𝑗 = 𝐽)
52 fvex 6158 . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . 14 (𝑚 = ((mapd‘𝐾)‘𝑊) → 𝑚 = 𝑀)
56 fveq1 6147 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑀 → (𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑀‘(𝑛‘{(2nd𝑥)})))
5756eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ↔ (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{})))
58 fveq1 6147 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑀 → (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})))
5958eqeq1d 2623 . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . 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 6567 . . . . . . . . . . . . . . . . 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 4077 . . . . . . . . . . . . . . . 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 4705 . . . . . . . . . . . . . . 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 2684 . . . . . . . . . . . . . 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 3452 . . . . . . . . . . . 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 1060 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑑 = 𝐷)
68 xpeq2 5089 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → (𝑣 × 𝑑) = (𝑣 × 𝐷))
6968xpeq1d 5098 . . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑐 = 𝐶)
7271fveq2d 6152 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (0g𝑐) = (0g𝐶))
73 hdmap1fval.q . . . . . . . . . . . . . . . 16 𝑄 = (0g𝐶)
7472, 73syl6eqr 2673 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (0g𝑐) = 𝑄)
75 simp3 1061 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → 𝑗 = 𝐽)
7675fveq1d 6150 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑗‘{}) = (𝐽‘{}))
7776eqeq2d 2631 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ↔ (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{})))
7871fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (-g𝑐) = (-g𝐶))
79 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (-g𝐶)
8078, 79syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (-g𝑐) = 𝑅)
8180oveqd 6621 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → ((2nd ‘(1st𝑥))(-g𝑐)) = ((2nd ‘(1st𝑥))𝑅))
8281sneqd 4160 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → {((2nd ‘(1st𝑥))(-g𝑐))} = {((2nd ‘(1st𝑥))𝑅)})
8375, 82fveq12d 6154 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑑 = 𝐷𝑗 = 𝐽) → (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))}) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))
8483eqeq2d 2631 . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . 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 6568 . . . . . . . . . . . . . . 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 4078 . . . . . . . . . . . . . 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 4693 . . . . . . . . . . . . 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 2684 . . . . . . . . . . . 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 272 . . . . . . . . . . 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 1323 . . . . . . . . . 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 3489 . . . . . . . . 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 1060 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑣 = 𝑉)
9493xpeq1d 5098 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑣 × 𝐷) = (𝑉 × 𝐷))
9594, 93xpeq12d 5100 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑣 × 𝐷) × 𝑣) = ((𝑉 × 𝐷) × 𝑉))
96 simp1 1059 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑢 = 𝑈)
9796fveq2d 6152 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (0g𝑢) = (0g𝑈))
98 hdmap1fval.o . . . . . . . . . . . . . 14 0 = (0g𝑈)
9997, 98syl6eqr 2673 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (0g𝑢) = 0 )
10099eqeq2d 2631 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((2nd𝑥) = (0g𝑢) ↔ (2nd𝑥) = 0 ))
101 simp3 1061 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → 𝑛 = 𝑁)
102101fveq1d 6150 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑛‘{(2nd𝑥)}) = (𝑁‘{(2nd𝑥)}))
103102fveq2d 6152 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑀‘(𝑛‘{(2nd𝑥)})) = (𝑀‘(𝑁‘{(2nd𝑥)})))
104103eqeq1d 2623 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑀‘(𝑛‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{})))
10596fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (-g𝑢) = (-g𝑈))
106 hdmap1fval.s . . . . . . . . . . . . . . . . . . . 20 = (-g𝑈)
107105, 106syl6eqr 2673 . . . . . . . . . . . . . . . . . . 19 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (-g𝑢) = )
108107oveqd 6621 . . . . . . . . . . . . . . . . . 18 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥)) = ((1st ‘(1st𝑥)) (2nd𝑥)))
109108sneqd 4160 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → {((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))} = {((1st ‘(1st𝑥)) (2nd𝑥))})
110101, 109fveq12d 6154 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))}) = (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}))
111110fveq2d 6152 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → (𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})))
112111eqeq1d 2623 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑣 = 𝑉𝑛 = 𝑁) → ((𝑀‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))
113104, 112anbi12d 746 . . . . . . . . . . . . 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)}))))
114113riotabidv 6567 . . . . . . . . . . . 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)}))))
115100, 114ifbieq2d 4083 . . . . . . . . . . 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)})))))
11695, 115mpteq12dv 4693 . . . . . . . . . 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)}))))))
117116eleq2d 2684 . . . . . . . . 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)})))))))
11892, 117syl5bb 272 . . . . . . . 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)})))))))
11923, 29, 34, 118syl3anc 1323 . . . . . . 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)})))))))
12017, 18, 19, 119sbc3ie 3489 . . . . . 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)}))))))
12116, 120syl6bb 276 . . . . 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)})))))))
122121abbi1dv 2740 . . . 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)}))))))
123 eqid 2621 . . . 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))})))))})
124 fvex 6158 . . . . . . . 8 (Base‘𝑈) ∈ V
12528, 124eqeltri 2694 . . . . . . 7 𝑉 ∈ V
126 fvex 6158 . . . . . . . 8 (Base‘𝐶) ∈ V
12744, 126eqeltri 2694 . . . . . . 7 𝐷 ∈ V
128125, 127xpex 6915 . . . . . 6 (𝑉 × 𝐷) ∈ V
129128, 125xpex 6915 . . . . 5 ((𝑉 × 𝐷) × 𝑉) ∈ V
130129mptex 6440 . . . 4 (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))) ∈ V
131122, 123, 130fvmpt 6239 . . 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)}))))))
1326, 131sylan9eq 2675 . 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)}))))))
1331, 132syl 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  [wsbc 3417  ifcif 4058  {csn 4148  cmpt 4673   × cxp 5072  cfv 5847  crio 6564  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Basecbs 15781  0gc0g 16021  -gcsg 17345  LSpanclspn 18890  LHypclh 34747  DVecHcdvh 35844  LCDualclcd 36352  mapdcmpd 36390  HDMap1chdma1 36558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-hdmap1 36560
This theorem is referenced by:  hdmap1vallem  36564
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