Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmapfval Structured version   Visualization version   GIF version

Theorem hdmapfval 37436
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 37435 . . . . 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 6231 . . . 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 2697 . . 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 6229 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
87reseq2d 5428 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ ((LTrn‘𝐾)‘𝑤)) = ( I ↾ ((LTrn‘𝐾)‘𝑊)))
98opeq2d 4440 . . . . . . 7 (𝑤 = 𝑊 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)
10 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
11 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap1‘𝐾)‘𝑤) = ((HDMap1‘𝐾)‘𝑊))
12 fveq2 6229 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((LCDual‘𝐾)‘𝑤) = ((LCDual‘𝐾)‘𝑊))
1312fveq2d 6233 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Base‘((LCDual‘𝐾)‘𝑤)) = (Base‘((LCDual‘𝐾)‘𝑊)))
14 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑊 → ((HVMap‘𝐾)‘𝑤) = ((HVMap‘𝐾)‘𝑊))
1514fveq1d 6231 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑊 → (((HVMap‘𝐾)‘𝑤)‘𝑒) = (((HVMap‘𝐾)‘𝑊)‘𝑒))
1615oteq2d 4446 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑊 → ⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩ = ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩)
1716fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑊 → (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩) = (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩))
1817oteq2d 4446 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)
1918fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
2019eqeq2d 2661 . . . . . . . . . . . . . . 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⟩)))
2120imbi2d 329 . . . . . . . . . . . . . 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⟩))))
2221ralbidv 3015 . . . . . . . . . . . . 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⟩))))
2313, 22riotaeqbidv 6654 . . . . . . . . . . . 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⟩))))
2423mpteq2dv 4778 . . . . . . . . . . 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⟩)))))
2524eleq2d 2716 . . . . . . . . . 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⟩))))))
2611, 25sbceqbid 3475 . . . . . . . . 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⟩))))))
2726sbcbidv 3523 . . . . . . . 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⟩))))))
2810, 27sbceqbid 3475 . . . . . . 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⟩))))))
299, 28sbceqbid 3475 . . . . . 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⟩))))))
30 opex 4962 . . . . . . 7 ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ V
31 fvex 6239 . . . . . . 7 ((DVecH‘𝐾)‘𝑊) ∈ V
32 fvex 6239 . . . . . . 7 (Base‘𝑢) ∈ V
33 simp1 1081 . . . . . . . . 9 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)
34 hdmapfval.e . . . . . . . . 9 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3533, 34syl6eqr 2703 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑒 = 𝐸)
36 simp2 1082 . . . . . . . . 9 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = ((DVecH‘𝐾)‘𝑊))
37 hdmapfval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
3836, 37syl6eqr 2703 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
39 simp3 1083 . . . . . . . . . 10 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = (Base‘𝑢))
4038fveq2d 6233 . . . . . . . . . 10 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → (Base‘𝑢) = (Base‘𝑈))
4139, 40eqtrd 2685 . . . . . . . . 9 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = (Base‘𝑈))
42 hdmapfval.v . . . . . . . . 9 𝑉 = (Base‘𝑈)
4341, 42syl6eqr 2703 . . . . . . . 8 ((𝑒 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∧ 𝑢 = ((DVecH‘𝐾)‘𝑊) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
44 fvex 6239 . . . . . . . . . 10 ((HDMap1‘𝐾)‘𝑊) ∈ V
45 id 22 . . . . . . . . . . . 12 (𝑖 = ((HDMap1‘𝐾)‘𝑊) → 𝑖 = ((HDMap1‘𝐾)‘𝑊))
46 hdmapfval.i . . . . . . . . . . . 12 𝐼 = ((HDMap1‘𝐾)‘𝑊)
4745, 46syl6eqr 2703 . . . . . . . . . . 11 (𝑖 = ((HDMap1‘𝐾)‘𝑊) → 𝑖 = 𝐼)
48 fveq1 6228 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
49 fveq1 6228 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩) = (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩))
5049oteq2d 4446 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩)
5150fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (𝐼‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
5248, 51eqtrd 2685 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩))
5352eqeq2d 2661 . . . . . . . . . . . . . . . 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⟩)))
5453imbi2d 329 . . . . . . . . . . . . . . 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⟩))))
5554ralbidv 3015 . . . . . . . . . . . . . 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⟩))))
5655riotabidv 6653 . . . . . . . . . . . . 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⟩))))
5756mpteq2dv 4778 . . . . . . . . . . . 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⟩)))))
5857eleq2d 2716 . . . . . . . . . . 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⟩))))))
5947, 58syl 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⟩))))))
6044, 59sbcie 3503 . . . . . . . . 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⟩)))))
61 simp3 1083 . . . . . . . . . . 11 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
62 hdmapfval.d . . . . . . . . . . . . . 14 𝐷 = (Base‘𝐶)
63 hdmapfval.c . . . . . . . . . . . . . . 15 𝐶 = ((LCDual‘𝐾)‘𝑊)
6463fveq2i 6232 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘((LCDual‘𝐾)‘𝑊))
6562, 64eqtr2i 2674 . . . . . . . . . . . . 13 (Base‘((LCDual‘𝐾)‘𝑊)) = 𝐷
6665a1i 11 . . . . . . . . . . . 12 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (Base‘((LCDual‘𝐾)‘𝑊)) = 𝐷)
67 simp2 1082 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
6867fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (LSpan‘𝑢) = (LSpan‘𝑈))
69 hdmapfval.n . . . . . . . . . . . . . . . . . . 19 𝑁 = (LSpan‘𝑈)
7068, 69syl6eqr 2703 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (LSpan‘𝑢) = 𝑁)
71 simp1 1081 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → 𝑒 = 𝐸)
7271sneqd 4222 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → {𝑒} = {𝐸})
7370, 72fveq12d 6235 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((LSpan‘𝑢)‘{𝑒}) = (𝑁‘{𝐸}))
7470fveq1d 6231 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ((LSpan‘𝑢)‘{𝑡}) = (𝑁‘{𝑡}))
7573, 74uneq12d 3801 . . . . . . . . . . . . . . . 16 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) = ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})))
7675eleq2d 2716 . . . . . . . . . . . . . . 15 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) ↔ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡}))))
7776notbid 307 . . . . . . . . . . . . . 14 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) ↔ ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡}))))
7871oteq1d 4445 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩)
7971fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((HVMap‘𝐾)‘𝑊)‘𝑒) = (((HVMap‘𝐾)‘𝑊)‘𝐸))
80 hdmapfval.j . . . . . . . . . . . . . . . . . . . . . 22 𝐽 = ((HVMap‘𝐾)‘𝑊)
8180fveq1i 6230 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝐸) = (((HVMap‘𝐾)‘𝑊)‘𝐸)
8279, 81syl6eqr 2703 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (((HVMap‘𝐾)‘𝑊)‘𝑒) = (𝐽𝐸))
8382oteq2d 4446 . . . . . . . . . . . . . . . . . . 19 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝐸, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (𝐽𝐸), 𝑧⟩)
8478, 83eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩ = ⟨𝐸, (𝐽𝐸), 𝑧⟩)
8584fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩))
8685oteq2d 4446 . . . . . . . . . . . . . . . 16 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → ⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)
8786fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))
8887eqeq2d 2661 . . . . . . . . . . . . . 14 ((𝑒 = 𝐸𝑢 = 𝑈𝑣 = 𝑉) → (𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑒, (((HVMap‘𝐾)‘𝑊)‘𝑒), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))
8977, 88imbi12d 333 . . . . . . . . . . . . 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⟩))))
9061, 89raleqbidv 3182 . . . . . . . . . . . 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⟩))))
9166, 90riotaeqbidv 6654 . . . . . . . . . . 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⟩))))
9261, 91mpteq12dv 4766 . . . . . . . . . 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⟩)))))
9392eleq2d 2716 . . . . . . . . 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⟩))))))
9460, 93syl5bb 272 . . . . . . . 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⟩))))))
9535, 38, 43, 94syl3anc 1366 . . . . . . 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⟩))))))
9630, 31, 32, 95sbc3ie 3540 . . . . . 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⟩)))))
9729, 96syl6bb 276 . . . . 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⟩))))))
9897abbi1dv 2772 . . . 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⟩)))))
99 eqid 2651 . . . 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⟩))))})
100 fvex 6239 . . . . . 6 (Base‘𝑈) ∈ V
10142, 100eqeltri 2726 . . . . 5 𝑉 ∈ V
102101mptex 6527 . . . 4 (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))) ∈ V
10398, 99, 102fvmpt 6321 . . 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⟩)))))
1046, 103sylan9eq 2705 . 2 ((𝐾𝐴𝑊𝐻) → 𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
1051, 104syl 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  [wsbc 3468  cun 3605  {csn 4210  cop 4216  cotp 4218  cmpt 4762   I cid 5052  cres 5145  cfv 5926  crio 6650  Basecbs 15904  LSpanclspn 19019  LHypclh 35588  LTrncltrn 35705  DVecHcdvh 36684  LCDualclcd 37192  HVMapchvm 37362  HDMap1chdma1 37398  HDMapchdma 37399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-hdmap 37401
This theorem is referenced by:  hdmapval  37437  hdmapfnN  37438
  Copyright terms: Public domain W3C validator