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Theorem hdmapfval 41211
Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h 𝐻 = (LHyp‘𝐾)
hdmapfval.e 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapfval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapfval.v 𝑉 = (Base‘𝑈)
hdmapfval.n 𝑁 = (LSpan‘𝑈)
hdmapfval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapfval.d 𝐷 = (Base‘𝐶)
hdmapfval.j 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmapfval.i 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmapfval.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapfval.k (𝜑 → (𝐾𝐴𝑊𝐻))
Assertion
Ref Expression
hdmapfval (𝜑𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
Distinct variable groups:   𝑦,𝑡,𝑧,𝐾   𝑦,𝐷   𝑡,𝐸,𝑦,𝑧   𝑡,𝐼,𝑦,𝑧   𝑡,𝑈,𝑦,𝑧   𝑡,𝑉,𝑦,𝑧   𝑡,𝑊,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑡)   𝐴(𝑦,𝑧,𝑡)   𝐶(𝑦,𝑧,𝑡)   𝐷(𝑧,𝑡)   𝑆(𝑦,𝑧,𝑡)   𝐻(𝑦,𝑧,𝑡)   𝐽(𝑦,𝑧,𝑡)   𝑁(𝑦,𝑧,𝑡)

Proof of Theorem hdmapfval
Dummy variables 𝑤 𝑒 𝑎 𝑖 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmapfval.k . 2 (𝜑 → (𝐾𝐴𝑊𝐻))
2 hdmapfval.s . . . 4 𝑆 = ((HDMap‘𝐾)‘𝑊)
3 hdmapval.h . . . . . 6 𝐻 = (LHyp‘𝐾)
43hdmapffval 41210 . . . . 5 (𝐾𝐴 → (HDMap‘𝐾) = (𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}))
54fveq1d 6887 . . . 4 (𝐾𝐴 → ((HDMap‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))})‘𝑊))
62, 5eqtrid 2778 . . 3 (𝐾𝐴𝑆 = ((𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))})‘𝑊))
7 fveq2 6885 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
87reseq2d 5975 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ ((LTrn‘𝐾)‘𝑤)) = ( I ↾ ((LTrn‘𝐾)‘𝑊)))
98opeq2d 4875 . . . . . . 7 (𝑤 = 𝑊 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)
10 fveq2 6885 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
11 fveq2 6885 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap1‘𝐾)‘𝑤) = ((HDMap1‘𝐾)‘𝑊))
12 2fveq3 6890 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Base‘((LCDual‘𝐾)‘𝑤)) = (Base‘((LCDual‘𝐾)‘𝑊)))
13 fveq2 6885 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑊 → ((HVMap‘𝐾)‘𝑤) = ((HVMap‘𝐾)‘𝑊))
1413fveq1d 6887 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑊 → (((HVMap‘𝐾)‘𝑤)‘𝑒) = (((HVMap‘𝐾)‘𝑊)‘𝑒))
1514oteq2d 4881 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑊 → ⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩ = ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩)
1615fveq2d 6889 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑊 → (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩) = (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩))
1716oteq2d 4881 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)
1817fveq2d 6889 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
1918eqeq2d 2737 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))
2019imbi2d 340 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)) ↔ (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))
2120ralbidv 3171 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)) ↔ ∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))
2212, 21riotaeqbidv 7364 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))) = (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))
2322mpteq2dv 5243 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) = (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))))
2423eleq2d 2813 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
2511, 24sbceqbid 3779 . . . . . . . . 9 (𝑤 = 𝑊 → ([((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ [((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
2625sbcbidv 3831 . . . . . . . 8 (𝑤 = 𝑊 → ([(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ [(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
2710, 26sbceqbid 3779 . . . . . . 7 (𝑤 = 𝑊 → ([((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ [((DVecH‘𝐾)‘𝑊) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
289, 27sbceqbid 3779 . . . . . 6 (𝑤 = 𝑊 → ([⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ [⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ / 𝑒][((DVecH‘𝐾)‘𝑊) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
29 opex 5457 . . . . . . 7 ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ V
30 fvex 6898 . . . . . . 7 ((DVecH‘𝐾)‘𝑊) ∈ V
31 fvex 6898 . . . . . . 7 (Base‘𝑢) ∈ V
32 simp1 1133 . . . . . . . . 9 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)
33 hdmapfval.e . . . . . . . . 9 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3432, 33eqtr4di 2784 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑒 = 𝐸)
35 simp2 1134 . . . . . . . . 9 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = ((DVecH‘𝐾)‘𝑊))
36 hdmapfval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
3735, 36eqtr4di 2784 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
38 simp3 1135 . . . . . . . . . 10 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = (Base‘𝑢))
3937fveq2d 6889 . . . . . . . . . 10 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → (Base‘𝑢) = (Base‘𝑈))
4038, 39eqtrd 2766 . . . . . . . . 9 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = (Base‘𝑈))
41 hdmapfval.v . . . . . . . . 9 𝑉 = (Base‘𝑈)
4240, 41eqtr4di 2784 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
43 fvex 6898 . . . . . . . . . 10 ((HDMap1‘𝐾)‘𝑊) ∈ V
44 id 22 . . . . . . . . . . . 12 (𝑖 = ((HDMap1‘𝐾)‘𝑊) → 𝑖 = ((HDMap1‘𝐾)‘𝑊))
45 hdmapfval.i . . . . . . . . . . . 12 𝐼 = ((HDMap1‘𝐾)‘𝑊)
4644, 45eqtr4di 2784 . . . . . . . . . . 11 (𝑖 = ((HDMap1‘𝐾)‘𝑊) → 𝑖 = 𝐼)
47 fveq1 6884 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
48 fveq1 6884 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩) = (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩))
4948oteq2d 4881 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)
5049fveq2d 6889 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (𝐼‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
5147, 50eqtrd 2766 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
5251eqeq2d 2737 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐼 → (𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))
5352imbi2d 340 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → ((¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)) ↔ (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))
5453ralbidv 3171 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)) ↔ ∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))
5554riotabidv 7363 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))) = (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))
5655mpteq2dv 5243 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) = (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))))
5756eleq2d 2813 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
5846, 57syl 17 . . . . . . . . . 10 (𝑖 = ((HDMap1‘𝐾)‘𝑊) → (𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))))))
5943, 58sbcie 3815 . . . . . . . . 9 ([((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))))
60 simp3 1135 . . . . . . . . . . 11 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
61 hdmapfval.d . . . . . . . . . . . . . 14 𝐷 = (Base‘𝐶)
62 hdmapfval.c . . . . . . . . . . . . . . 15 𝐶 = ((LCDual‘𝐾)‘𝑊)
6362fveq2i 6888 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘((LCDual‘𝐾)‘𝑊))
6461, 63eqtr2i 2755 . . . . . . . . . . . . 13 (Base‘((LCDual‘𝐾)‘𝑊)) = 𝐷
6564a1i 11 . . . . . . . . . . . 12 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (Base‘((LCDual‘𝐾)‘𝑊)) = 𝐷)
66 simp2 1134 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
6766fveq2d 6889 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (LSpan‘𝑢) = (LSpan‘𝑈))
68 hdmapfval.n . . . . . . . . . . . . . . . . . . 19 𝑁 = (LSpan‘𝑈)
6967, 68eqtr4di 2784 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (LSpan‘𝑢) = 𝑁)
70 simp1 1133 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑒 = 𝐸)
7170sneqd 4635 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → {𝑒} = {𝐸})
7269, 71fveq12d 6892 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((LSpan‘𝑢)‘{𝑒}) = (𝑁‘{𝐸}))
7369fveq1d 6887 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((LSpan‘𝑢)‘{𝑡}) = (𝑁‘{𝑡}))
7472, 73uneq12d 4159 . . . . . . . . . . . . . . . 16 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) = ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})))
7574eleq2d 2813 . . . . . . . . . . . . . . 15 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) ↔ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡}))))
7675notbid 318 . . . . . . . . . . . . . 14 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) ↔ ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡}))))
7770oteq1d 4880 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩)
7870fveq2d 6889 . . . . . . . . . . . . . . . . . . . . 21 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((HVMap‘𝐾)‘𝑊)‘𝑒) = (((HVMap‘𝐾)‘𝑊)‘𝐸))
79 hdmapfval.j . . . . . . . . . . . . . . . . . . . . . 22 𝐽 = ((HVMap‘𝐾)‘𝑊)
8079fveq1i 6886 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝐸) = (((HVMap‘𝐾)‘𝑊)‘𝐸)
8178, 80eqtr4di 2784 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((HVMap‘𝐾)‘𝑊)‘𝑒) = (𝐽𝐸))
8281oteq2d 4881 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝐸, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (𝐽𝐸), 𝑧⟩)
8377, 82eqtrd 2766 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (𝐽𝐸), 𝑧⟩)
8483fveq2d 6889 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩))
8584oteq2d 4881 . . . . . . . . . . . . . . . 16 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)
8685fveq2d 6889 . . . . . . . . . . . . . . 15 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))
8786eqeq2d 2737 . . . . . . . . . . . . . 14 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))
8876, 87imbi12d 344 . . . . . . . . . . . . 13 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)) ↔ (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))
8960, 88raleqbidv 3336 . . . . . . . . . . . 12 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)) ↔ ∀𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))
9065, 89riotaeqbidv 7364 . . . . . . . . . . 11 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))
9160, 90mpteq12dv 5232 . . . . . . . . . 10 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
9291eleq2d 2813 . . . . . . . . 9 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))))
9359, 92bitrid 283 . . . . . . . 8 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ([((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))))
9434, 37, 42, 93syl3anc 1368 . . . . . . 7 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → ([((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))))
9529, 30, 31, 94sbc3ie 3858 . . . . . 6 ([⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ / 𝑒][((DVecH‘𝐾)‘𝑊) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑊) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
9628, 95bitrdi 287 . . . . 5 (𝑤 = 𝑊 → ([⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))) ↔ 𝑎 ∈ (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))))
9796eqabcdv 2862 . . . 4 (𝑤 = 𝑊 → {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))} = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
98 eqid 2726 . . . 4 (𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}) = (𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))})
9997, 98, 41mptfvmpt 7225 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))})‘𝑊) = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
1006, 99sylan9eq 2786 . 2 ((𝐾𝐴𝑊𝐻) → 𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
1011, 100syl 17 1 (𝜑𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  {cab 2703  wral 3055  [wsbc 3772  cun 3941  {csn 4623  cop 4629  cotp 4631  cmpt 5224   I cid 5566  cres 5671  cfv 6537  crio 7360  Basecbs 17153  LSpanclspn 20818  LHypclh 39368  LTrncltrn 39485  DVecHcdvh 40462  LCDualclcd 40970  HVMapchvm 41140  HDMap1chdma1 41175  HDMapchdma 41176
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-hdmap 41178
This theorem is referenced by:  hdmapval  41212  hdmapfnN  41213
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