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Theorem hdmapfval 38830
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 38829 . . . . 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 6669 . . . 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, 5syl5eq 2873 . . 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 6667 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
87reseq2d 5852 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ ((LTrn‘𝐾)‘𝑤)) = ( I ↾ ((LTrn‘𝐾)‘𝑊)))
98opeq2d 4809 . . . . . . 7 (𝑤 = 𝑊 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)
10 fveq2 6667 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
11 fveq2 6667 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap1‘𝐾)‘𝑤) = ((HDMap1‘𝐾)‘𝑊))
12 2fveq3 6672 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Base‘((LCDual‘𝐾)‘𝑤)) = (Base‘((LCDual‘𝐾)‘𝑊)))
13 fveq2 6667 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑊 → ((HVMap‘𝐾)‘𝑤) = ((HVMap‘𝐾)‘𝑊))
1413fveq1d 6669 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑊 → (((HVMap‘𝐾)‘𝑤)‘𝑒) = (((HVMap‘𝐾)‘𝑊)‘𝑒))
1514oteq2d 4815 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑊 → ⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩ = ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩)
1615fveq2d 6671 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑊 → (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩) = (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩))
1716oteq2d 4815 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)
1817fveq2d 6671 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
1918eqeq2d 2837 . . . . . . . . . . . . . . 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 342 . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . 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 7109 . . . . . . . . . . . 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 5159 . . . . . . . . . . 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 2903 . . . . . . . . . 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 3783 . . . . . . . . 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 3783 . . . . . . 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 3783 . . . . . 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 5353 . . . . . . 7 ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ V
30 fvex 6680 . . . . . . 7 ((DVecH‘𝐾)‘𝑊) ∈ V
31 fvex 6680 . . . . . . 7 (Base‘𝑢) ∈ V
32 simp1 1130 . . . . . . . . 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, 33syl6eqr 2879 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑒 = 𝐸)
35 simp2 1131 . . . . . . . . 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, 36syl6eqr 2879 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
38 simp3 1132 . . . . . . . . . 10 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = (Base‘𝑢))
3937fveq2d 6671 . . . . . . . . . 10 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → (Base‘𝑢) = (Base‘𝑈))
4038, 39eqtrd 2861 . . . . . . . . 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, 41syl6eqr 2879 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
43 fvex 6680 . . . . . . . . . 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, 45syl6eqr 2879 . . . . . . . . . . 11 (𝑖 = ((HDMap1‘𝐾)‘𝑊) → 𝑖 = 𝐼)
47 fveq1 6666 . . . . . . . . . . . . . . . . . 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 6666 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩) = (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩))
4948oteq2d 4815 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)
5049fveq2d 6671 . . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
5251eqeq2d 2837 . . . . . . . . . . . . . . . 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 342 . . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . . 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 7108 . . . . . . . . . . . . 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 5159 . . . . . . . . . . . 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 2903 . . . . . . . . . . 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 3816 . . . . . . . . 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 1132 . . . . . . . . . . 11 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
61 hdmapfval.d . . . . . . . . . . . . . 14 𝐷 = (Base‘𝐶)
62 hdmapfval.c . . . . . . . . . . . . . . 15 𝐶 = ((LCDual‘𝐾)‘𝑊)
6362fveq2i 6670 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘((LCDual‘𝐾)‘𝑊))
6461, 63eqtr2i 2850 . . . . . . . . . . . . 13 (Base‘((LCDual‘𝐾)‘𝑊)) = 𝐷
6564a1i 11 . . . . . . . . . . . 12 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (Base‘((LCDual‘𝐾)‘𝑊)) = 𝐷)
66 simp2 1131 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
6766fveq2d 6671 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (LSpan‘𝑢) = (LSpan‘𝑈))
68 hdmapfval.n . . . . . . . . . . . . . . . . . . 19 𝑁 = (LSpan‘𝑈)
6967, 68syl6eqr 2879 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (LSpan‘𝑢) = 𝑁)
70 simp1 1130 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑒 = 𝐸)
7170sneqd 4576 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → {𝑒} = {𝐸})
7269, 71fveq12d 6674 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((LSpan‘𝑢)‘{𝑒}) = (𝑁‘{𝐸}))
7369fveq1d 6669 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((LSpan‘𝑢)‘{𝑡}) = (𝑁‘{𝑡}))
7472, 73uneq12d 4144 . . . . . . . . . . . . . . . 16 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) = ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})))
7574eleq2d 2903 . . . . . . . . . . . . . . 15 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) ↔ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡}))))
7675notbid 319 . . . . . . . . . . . . . 14 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) ↔ ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡}))))
7770oteq1d 4814 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩)
7870fveq2d 6671 . . . . . . . . . . . . . . . . . . . . 21 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((HVMap‘𝐾)‘𝑊)‘𝑒) = (((HVMap‘𝐾)‘𝑊)‘𝐸))
79 hdmapfval.j . . . . . . . . . . . . . . . . . . . . . 22 𝐽 = ((HVMap‘𝐾)‘𝑊)
8079fveq1i 6668 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝐸) = (((HVMap‘𝐾)‘𝑊)‘𝐸)
8178, 80syl6eqr 2879 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((HVMap‘𝐾)‘𝑊)‘𝑒) = (𝐽𝐸))
8281oteq2d 4815 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝐸, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (𝐽𝐸), 𝑧⟩)
8377, 82eqtrd 2861 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (𝐽𝐸), 𝑧⟩)
8483fveq2d 6671 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩))
8584oteq2d 4815 . . . . . . . . . . . . . . . 16 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)
8685fveq2d 6671 . . . . . . . . . . . . . . 15 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))
8786eqeq2d 2837 . . . . . . . . . . . . . 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 346 . . . . . . . . . . . . 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 3407 . . . . . . . . . . . 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 7109 . . . . . . . . . . 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 5148 . . . . . . . . . 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 2903 . . . . . . . . 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, 92syl5bb 284 . . . . . . . 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 1365 . . . . . . 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 3856 . . . . . 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, 95syl6bb 288 . . . . 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⟩))))))
9796abbi1dv 2957 . . . 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 2826 . . . 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 6985 . . 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 2881 . 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {cab 2804  wral 3143  [wsbc 3776  cun 3938  {csn 4564  cop 4570  cotp 4572  cmpt 5143   I cid 5458  cres 5556  cfv 6352  crio 7105  Basecbs 16473  LSpanclspn 19663  LHypclh 36987  LTrncltrn 37104  DVecHcdvh 38081  LCDualclcd 38589  HVMapchvm 38759  HDMap1chdma1 38794  HDMapchdma 38795
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-ot 4573  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-hdmap 38797
This theorem is referenced by:  hdmapval  38831  hdmapfnN  38832
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