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Theorem hdmapffval 41191
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 3485 . 2 (𝐾𝑋𝐾 ∈ V)
2 fveq2 6882 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 hdmapval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2782 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6882 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
65reseq2d 5972 . . . . . . 7 (𝑘 = 𝐾 → ( I ↾ (Base‘𝑘)) = ( I ↾ (Base‘𝐾)))
7 fveq2 6882 . . . . . . . . 9 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
87fveq1d 6884 . . . . . . . 8 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
98reseq2d 5972 . . . . . . 7 (𝑘 = 𝐾 → ( I ↾ ((LTrn‘𝑘)‘𝑤)) = ( I ↾ ((LTrn‘𝐾)‘𝑤)))
106, 9opeq12d 4874 . . . . . 6 (𝑘 = 𝐾 → ⟨( I ↾ (Base‘𝑘)), ( I ↾ ((LTrn‘𝑘)‘𝑤))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩)
11 fveq2 6882 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
1211fveq1d 6884 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
13 fveq2 6882 . . . . . . . . . 10 (𝑘 = 𝐾 → (HDMap1‘𝑘) = (HDMap1‘𝐾))
1413fveq1d 6884 . . . . . . . . 9 (𝑘 = 𝐾 → ((HDMap1‘𝑘)‘𝑤) = ((HDMap1‘𝐾)‘𝑤))
15 fveq2 6882 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (LCDual‘𝑘) = (LCDual‘𝐾))
1615fveq1d 6884 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → ((LCDual‘𝑘)‘𝑤) = ((LCDual‘𝐾)‘𝑤))
1716fveq2d 6886 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (Base‘((LCDual‘𝑘)‘𝑤)) = (Base‘((LCDual‘𝐾)‘𝑤)))
18 fveq2 6882 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝐾 → (HVMap‘𝑘) = (HVMap‘𝐾))
1918fveq1d 6884 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐾 → ((HVMap‘𝑘)‘𝑤) = ((HVMap‘𝐾)‘𝑤))
2019fveq1d 6884 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝐾 → (((HVMap‘𝑘)‘𝑤)‘𝑒) = (((HVMap‘𝐾)‘𝑤)‘𝑒))
2120oteq2d 4879 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝐾 → ⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩ = ⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩)
2221fveq2d 6886 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝐾 → (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩) = (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩))
2322oteq2d 4879 . . . . . . . . . . . . . . . 16 (𝑘 = 𝐾 → ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩)
2423fveq2d 6886 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩) = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))
2524eqeq2d 2735 . . . . . . . . . . . . . 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 340 . . . . . . . . . . . . 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 3169 . . . . . . . . . . . 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 7361 . . . . . . . . . . 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 5241 . . . . . . . . . 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 3777 . . . . . . . 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 3829 . . . . . . 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 3777 . . . . . 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 3777 . . . . 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 2793 . . . 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 5230 . . 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 41159 . . 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 7222 . 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 2701  wral 3053  Vcvv 3466  [wsbc 3770  cun 3939  {csn 4621  cop 4627  cotp 4629  cmpt 5222   I cid 5564  cres 5669  cfv 6534  crio 7357  Basecbs 17145  LSpanclspn 20810  LHypclh 39349  LTrncltrn 39466  DVecHcdvh 40443  LCDualclcd 40951  HVMapchvm 41121  HDMap1chdma1 41156  HDMapchdma 41157
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-hdmap 41159
This theorem is referenced by:  hdmapfval  41192
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