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Theorem hdmapffval 41355
Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
Hypothesis
Ref Expression
hdmapval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
hdmapffval (𝐾𝑋 → (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⟩))))}))
Distinct variable groups:   𝑤,𝐻   𝑒,𝑎,𝑖,𝑡,𝑢,𝑣,𝑤,𝑦,𝑧,𝐾
Allowed substitution hints:   𝐻(𝑦,𝑧,𝑣,𝑢,𝑡,𝑒,𝑖,𝑎)   𝑋(𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑒,𝑖,𝑎)

Proof of Theorem hdmapffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3482 . 2 (𝐾𝑋𝐾 ∈ V)
2 fveq2 6892 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 hdmapval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2783 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6892 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
65reseq2d 5979 . . . . . . 7 (𝑘 = 𝐾 → ( I ↾ (Base‘𝑘)) = ( I ↾ (Base‘𝐾)))
7 fveq2 6892 . . . . . . . . 9 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
87fveq1d 6894 . . . . . . . 8 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
98reseq2d 5979 . . . . . . 7 (𝑘 = 𝐾 → ( I ↾ ((LTrn‘𝑘)‘𝑤)) = ( I ↾ ((LTrn‘𝐾)‘𝑤)))
106, 9opeq12d 4877 . . . . . 6 (𝑘 = 𝐾 → ⟨( I ↾ (Base‘𝑘)), ( I ↾ ((LTrn‘𝑘)‘𝑤))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩)
11 fveq2 6892 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
1211fveq1d 6894 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
13 fveq2 6892 . . . . . . . . . 10 (𝑘 = 𝐾 → (HDMap1‘𝑘) = (HDMap1‘𝐾))
1413fveq1d 6894 . . . . . . . . 9 (𝑘 = 𝐾 → ((HDMap1‘𝑘)‘𝑤) = ((HDMap1‘𝐾)‘𝑤))
15 fveq2 6892 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (LCDual‘𝑘) = (LCDual‘𝐾))
1615fveq1d 6894 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → ((LCDual‘𝑘)‘𝑤) = ((LCDual‘𝐾)‘𝑤))
1716fveq2d 6896 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (Base‘((LCDual‘𝑘)‘𝑤)) = (Base‘((LCDual‘𝐾)‘𝑤)))
18 fveq2 6892 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝐾 → (HVMap‘𝑘) = (HVMap‘𝐾))
1918fveq1d 6894 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐾 → ((HVMap‘𝑘)‘𝑤) = ((HVMap‘𝐾)‘𝑤))
2019fveq1d 6894 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝐾 → (((HVMap‘𝑘)‘𝑤)‘𝑒) = (((HVMap‘𝐾)‘𝑤)‘𝑒))
2120oteq2d 4882 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝐾 → ⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩ = ⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩)
2221fveq2d 6896 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝐾 → (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩) = (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩))
2322oteq2d 4882 . . . . . . . . . . . . . . . 16 (𝑘 = 𝐾 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)
2423fveq2d 6896 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))
2524eqeq2d 2736 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)))
2625imbi2d 339 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → ((¬ 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⟩))))
2726ralbidv 3168 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (∀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⟩))))
2817, 27riotaeqbidv 7375 . . . . . . . . . . 11 (𝑘 = 𝐾 → (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⟩))))
2928mpteq2dv 5245 . . . . . . . . . 10 (𝑘 = 𝐾 → (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⟩)))))
3029eleq2d 2811 . . . . . . . . 9 (𝑘 = 𝐾 → (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⟩))))))
3114, 30sbceqbid 3775 . . . . . . . 8 (𝑘 = 𝐾 → ([((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⟩))))))
3231sbcbidv 3828 . . . . . . 7 (𝑘 = 𝐾 → ([(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⟩))))))
3312, 32sbceqbid 3775 . . . . . 6 (𝑘 = 𝐾 → ([((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⟩))))))
3410, 33sbceqbid 3775 . . . . 5 (𝑘 = 𝐾 → ([⟨( 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⟩))))))
3534abbidv 2794 . . . 4 (𝑘 = 𝐾 → {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⟩))))} = {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⟩))))})
364, 35mpteq12dv 5234 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {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⟩))))}))
37 df-hdmap 41323 . . 3 HDMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {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⟩))))}))
3836, 37, 3mptfvmpt 7236 . 2 (𝐾 ∈ V → (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⟩))))}))
391, 38syl 17 1 (𝐾𝑋 → (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⟩))))}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  {cab 2702  wral 3051  Vcvv 3463  [wsbc 3768  cun 3937  {csn 4624  cop 4630  cotp 4632  cmpt 5226   I cid 5569  cres 5674  cfv 6543  crio 7371  Basecbs 17179  LSpanclspn 20859  LHypclh 39513  LTrncltrn 39630  DVecHcdvh 40607  LCDualclcd 41115  HVMapchvm 41285  HDMap1chdma1 41320  HDMapchdma 41321
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-ot 4633  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-hdmap 41323
This theorem is referenced by:  hdmapfval  41356
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