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Theorem hlhilset 37890
Description: The final Hilbert space constructed from a Hilbert lattice 𝐾 and an arbitrary hyperplane 𝑊 in 𝐾. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Hypotheses
Ref Expression
hlhilset.h 𝐻 = (LHyp‘𝐾)
hlhilset.l 𝐿 = ((HLHil‘𝐾)‘𝑊)
hlhilset.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hlhilset.v 𝑉 = (Base‘𝑈)
hlhilset.p + = (+g𝑈)
hlhilset.e 𝐸 = ((EDRing‘𝐾)‘𝑊)
hlhilset.g 𝐺 = ((HGMap‘𝐾)‘𝑊)
hlhilset.r 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)
hlhilset.t · = ( ·𝑠𝑈)
hlhilset.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hlhilset.i , = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥))
hlhilset.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
Assertion
Ref Expression
hlhilset (𝜑𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
Distinct variable groups:   𝑥,𝑦,𝐾   𝜑,𝑥,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   + (𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   · (𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   , (𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem hlhilset
Dummy variables 𝑤 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlhilset.l . 2 𝐿 = ((HLHil‘𝐾)‘𝑊)
2 hlhilset.k . . . . 5 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 elex 3365 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ V)
43adantr 472 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐾 ∈ V)
52, 4syl 17 . . . 4 (𝜑𝐾 ∈ V)
6 hlhilset.h . . . . . 6 𝐻 = (LHyp‘𝐾)
76fvexi 6389 . . . . 5 𝐻 ∈ V
87mptex 6679 . . . 4 (𝑤𝐻𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})) ∈ V
9 nfcv 2907 . . . . 5 𝑘𝐾
10 nfcv 2907 . . . . . 6 𝑘𝐻
11 nfcsb1v 3707 . . . . . 6 𝑘𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})
1210, 11nfmpt 4905 . . . . 5 𝑘(𝑤𝐻𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}))
13 fveq2 6375 . . . . . . 7 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
1413, 6syl6eqr 2817 . . . . . 6 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
15 csbeq1a 3700 . . . . . 6 (𝑘 = 𝐾((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = 𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}))
1614, 15mpteq12dv 4892 . . . . 5 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ ((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})) = (𝑤𝐻𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
17 df-hlhil 37889 . . . . 5 HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
189, 12, 16, 17fvmptf 6490 . . . 4 ((𝐾 ∈ V ∧ (𝑤𝐻𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})) ∈ V) → (HLHil‘𝐾) = (𝑤𝐻𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
195, 8, 18sylancl 580 . . 3 (𝜑 → (HLHil‘𝐾) = (𝑤𝐻𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
205adantr 472 . . . 4 ((𝜑𝑤 = 𝑊) → 𝐾 ∈ V)
21 fvexd 6390 . . . . 5 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) ∈ V)
22 fvexd 6390 . . . . . 6 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) ∈ V)
23 id 22 . . . . . . . . . 10 (𝑣 = (Base‘𝑢) → 𝑣 = (Base‘𝑢))
24 id 22 . . . . . . . . . . . . 13 (𝑢 = ((DVecH‘𝑘)‘𝑤) → 𝑢 = ((DVecH‘𝑘)‘𝑤))
25 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
2625fveq2d 6379 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (DVecH‘𝑘) = (DVecH‘𝐾))
27 simplr 785 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑤 = 𝑊)
2826, 27fveq12d 6382 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
29 hlhilset.u . . . . . . . . . . . . . 14 𝑈 = ((DVecH‘𝐾)‘𝑊)
3028, 29syl6eqr 2817 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = 𝑈)
3124, 30sylan9eqr 2821 . . . . . . . . . . . 12 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → 𝑢 = 𝑈)
3231fveq2d 6379 . . . . . . . . . . 11 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = (Base‘𝑈))
33 hlhilset.v . . . . . . . . . . 11 𝑉 = (Base‘𝑈)
3432, 33syl6eqr 2817 . . . . . . . . . 10 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = 𝑉)
3523, 34sylan9eqr 2821 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
3635opeq2d 4566 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝑉⟩)
3731adantr 472 . . . . . . . . . . 11 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
3837fveq2d 6379 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g𝑢) = (+g𝑈))
39 hlhilset.p . . . . . . . . . 10 + = (+g𝑈)
4038, 39syl6eqr 2817 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g𝑢) = + )
4140opeq2d 4566 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(+g‘ndx), (+g𝑢)⟩ = ⟨(+g‘ndx), + ⟩)
4225fveq2d 6379 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (EDRing‘𝑘) = (EDRing‘𝐾))
4342, 27fveq12d 6382 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
44 hlhilset.e . . . . . . . . . . . . 13 𝐸 = ((EDRing‘𝐾)‘𝑊)
4543, 44syl6eqr 2817 . . . . . . . . . . . 12 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = 𝐸)
4625fveq2d 6379 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HGMap‘𝑘) = (HGMap‘𝐾))
4746, 27fveq12d 6382 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = ((HGMap‘𝐾)‘𝑊))
48 hlhilset.g . . . . . . . . . . . . . 14 𝐺 = ((HGMap‘𝐾)‘𝑊)
4947, 48syl6eqr 2817 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = 𝐺)
5049opeq2d 4566 . . . . . . . . . . . 12 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩ = ⟨(*𝑟‘ndx), 𝐺⟩)
5145, 50oveq12d 6860 . . . . . . . . . . 11 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩))
52 hlhilset.r . . . . . . . . . . 11 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)
5351, 52syl6eqr 2817 . . . . . . . . . 10 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = 𝑅)
5453opeq2d 4566 . . . . . . . . 9 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5554ad2antrr 717 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5636, 41, 55tpeq123d 4438 . . . . . . 7 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → {⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} = {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩})
5737fveq2d 6379 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
58 hlhilset.t . . . . . . . . . 10 · = ( ·𝑠𝑈)
5957, 58syl6eqr 2817 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠𝑢) = · )
6059opeq2d 4566 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
6125fveq2d 6379 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HDMap‘𝑘) = (HDMap‘𝐾))
6261, 27fveq12d 6382 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
63 hlhilset.s . . . . . . . . . . . . . . 15 𝑆 = ((HDMap‘𝐾)‘𝑊)
6462, 63syl6eqr 2817 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
6564ad2antrr 717 . . . . . . . . . . . . 13 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
6665fveq1d 6377 . . . . . . . . . . . 12 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (((HDMap‘𝑘)‘𝑤)‘𝑦) = (𝑆𝑦))
6766fveq1d 6377 . . . . . . . . . . 11 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥) = ((𝑆𝑦)‘𝑥))
6835, 35, 67mpt2eq123dv 6915 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥)))
69 hlhilset.i . . . . . . . . . 10 , = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥))
7068, 69syl6eqr 2817 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = , )
7170opeq2d 4566 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩ = ⟨(·𝑖‘ndx), , ⟩)
7260, 71preq12d 4431 . . . . . . 7 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩} = {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})
7356, 72uneq12d 3930 . . . . . 6 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
7422, 73csbied 3718 . . . . 5 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
7521, 74csbied 3718 . . . 4 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
7620, 75csbied 3718 . . 3 ((𝜑𝑤 = 𝑊) → 𝐾 / 𝑘((DVecH‘𝑘)‘𝑤) / 𝑢(Base‘𝑢) / 𝑣({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
772simprd 489 . . 3 (𝜑𝑊𝐻)
78 tpex 7155 . . . . 5 {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∈ V
79 prex 5065 . . . . 5 {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
8078, 79unex 7154 . . . 4 ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V
8180a1i 11 . . 3 (𝜑 → ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V)
8219, 76, 77, 81fvmptd 6477 . 2 (𝜑 → ((HLHil‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
831, 82syl5eq 2811 1 (𝜑𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  csb 3691  cun 3730  {cpr 4336  {ctp 4338  cop 4340  cmpt 4888  cfv 6068  (class class class)co 6842  cmpt2 6844  ndxcnx 16127   sSet csts 16128  Basecbs 16130  +gcplusg 16214  *𝑟cstv 16216  Scalarcsca 16217   ·𝑠 cvsca 16218  ·𝑖cip 16219  HLchlt 35306  LHypclh 35940  EDRingcedring 36709  DVecHcdvh 37034  HDMapchdma 37748  HGMapchg 37839  HLHilchlh 37888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-hlhil 37889
This theorem is referenced by:  hlhilsca  37891  hlhilbase  37892  hlhilplus  37893  hlhilvsca  37903  hlhilip  37904
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