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Theorem dvhset 35388
 Description: The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvhset.h 𝐻 = (LHyp‘𝐾)
dvhset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhset.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhset.d 𝐷 = ((EDRing‘𝐾)‘𝑊)
dvhset.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
Assertion
Ref Expression
dvhset ((𝐾𝑋𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
Distinct variable groups:   𝑓,𝑔,𝐻   𝑓,,𝑠,𝐾,𝑔   𝑇,   𝑓,𝑊,𝑔,,𝑠   𝑓,𝑋,𝑔
Allowed substitution hints:   𝐷(𝑓,𝑔,,𝑠)   𝑇(𝑓,𝑔,𝑠)   𝑈(𝑓,𝑔,,𝑠)   𝐸(𝑓,𝑔,,𝑠)   𝐻(,𝑠)   𝑋(,𝑠)

Proof of Theorem dvhset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dvhset.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
2 dvhset.h . . . . 5 𝐻 = (LHyp‘𝐾)
32dvhfset 35387 . . . 4 (𝐾𝑋 → (DVecH‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
43fveq1d 6105 . . 3 (𝐾𝑋 → ((DVecH‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))‘𝑊))
51, 4syl5eq 2656 . 2 (𝐾𝑋𝑈 = ((𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))‘𝑊))
6 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7 dvhset.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
86, 7syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
9 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
10 dvhset.e . . . . . . . 8 𝐸 = ((TEndo‘𝐾)‘𝑊)
119, 10syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
128, 11xpeq12d 5064 . . . . . 6 (𝑤 = 𝑊 → (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) = (𝑇 × 𝐸))
1312opeq2d 4347 . . . . 5 (𝑤 = 𝑊 → ⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩ = ⟨(Base‘ndx), (𝑇 × 𝐸)⟩)
148mpteq1d 4666 . . . . . . . 8 (𝑤 = 𝑊 → ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))) = (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))))
1514opeq2d 4347 . . . . . . 7 (𝑤 = 𝑊 → ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩ = ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)
1612, 12, 15mpt2eq123dv 6615 . . . . . 6 (𝑤 = 𝑊 → (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩))
1716opeq2d 4347 . . . . 5 (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩ = ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩)
18 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
19 dvhset.d . . . . . . 7 𝐷 = ((EDRing‘𝐾)‘𝑊)
2018, 19syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = 𝐷)
2120opeq2d 4347 . . . . 5 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩ = ⟨(Scalar‘ndx), 𝐷⟩)
2213, 17, 21tpeq123d 4227 . . . 4 (𝑤 = 𝑊 → {⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} = {⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩})
23 eqidd 2611 . . . . . . 7 (𝑤 = 𝑊 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
2411, 12, 23mpt2eq123dv 6615 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
2524opeq2d 4347 . . . . 5 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩ = ⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩)
2625sneqd 4137 . . . 4 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩} = {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})
2722, 26uneq12d 3730 . . 3 (𝑤 = 𝑊 → ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
28 eqid 2610 . . 3 (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
29 tpex 6855 . . . 4 {⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∈ V
30 snex 4835 . . . 4 {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩} ∈ V
3129, 30unex 6854 . . 3 ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}) ∈ V
3227, 28, 31fvmpt 6191 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))‘𝑊) = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
335, 32sylan9eq 2664 1 ((𝐾𝑋𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {csn 4125  {ctp 4129  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ‘cfv 5804   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058  EDRingcedring 35059  DVecHcdvh 35385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-dvech 35386 This theorem is referenced by:  dvhsca  35389  dvhvbase  35394  dvhfvadd  35398  dvhfvsca  35407
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