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Theorem hlhilset 39229
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 3462 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ V)
43adantr 484 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐾 ∈ V)
52, 4syl 17 . . . 4 (𝜑𝐾 ∈ V)
6 hlhilset.h . . . . . 6 𝐻 = (LHyp‘𝐾)
76fvexi 6663 . . . . 5 𝐻 ∈ V
87mptex 6967 . . . 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 2958 . . . . 5 𝑘𝐾
10 nfcv 2958 . . . . . 6 𝑘𝐻
11 nfcsb1v 3855 . . . . . 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 5130 . . . . 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 6649 . . . . . . 7 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
1413, 6eqtr4di 2854 . . . . . 6 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
15 csbeq1a 3845 . . . . . 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 5118 . . . . 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 39228 . . . . 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 6770 . . . 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 589 . . 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 484 . . . 4 ((𝜑𝑤 = 𝑊) → 𝐾 ∈ V)
21 fvexd 6664 . . . . 5 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) ∈ V)
22 fvexd 6664 . . . . . 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 488 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
2625fveq2d 6653 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (DVecH‘𝑘) = (DVecH‘𝐾))
27 simplr 768 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑤 = 𝑊)
2826, 27fveq12d 6656 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
29 hlhilset.u . . . . . . . . . . . . . 14 𝑈 = ((DVecH‘𝐾)‘𝑊)
3028, 29eqtr4di 2854 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = 𝑈)
3124, 30sylan9eqr 2858 . . . . . . . . . . . 12 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → 𝑢 = 𝑈)
3231fveq2d 6653 . . . . . . . . . . 11 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = (Base‘𝑈))
33 hlhilset.v . . . . . . . . . . 11 𝑉 = (Base‘𝑈)
3432, 33eqtr4di 2854 . . . . . . . . . 10 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = 𝑉)
3523, 34sylan9eqr 2858 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
3635opeq2d 4775 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝑉⟩)
3731adantr 484 . . . . . . . . . . 11 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
3837fveq2d 6653 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g𝑢) = (+g𝑈))
39 hlhilset.p . . . . . . . . . 10 + = (+g𝑈)
4038, 39eqtr4di 2854 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g𝑢) = + )
4140opeq2d 4775 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(+g‘ndx), (+g𝑢)⟩ = ⟨(+g‘ndx), + ⟩)
4225fveq2d 6653 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (EDRing‘𝑘) = (EDRing‘𝐾))
4342, 27fveq12d 6656 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
44 hlhilset.e . . . . . . . . . . . . 13 𝐸 = ((EDRing‘𝐾)‘𝑊)
4543, 44eqtr4di 2854 . . . . . . . . . . . 12 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = 𝐸)
4625fveq2d 6653 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HGMap‘𝑘) = (HGMap‘𝐾))
4746, 27fveq12d 6656 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = ((HGMap‘𝐾)‘𝑊))
48 hlhilset.g . . . . . . . . . . . . . 14 𝐺 = ((HGMap‘𝐾)‘𝑊)
4947, 48eqtr4di 2854 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = 𝐺)
5049opeq2d 4775 . . . . . . . . . . . 12 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩ = ⟨(*𝑟‘ndx), 𝐺⟩)
5145, 50oveq12d 7157 . . . . . . . . . . 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, 52eqtr4di 2854 . . . . . . . . . 10 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = 𝑅)
5453opeq2d 4775 . . . . . . . . 9 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5554ad2antrr 725 . . . . . . . 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 4647 . . . . . . 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 6653 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
58 hlhilset.t . . . . . . . . . 10 · = ( ·𝑠𝑈)
5957, 58eqtr4di 2854 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠𝑢) = · )
6059opeq2d 4775 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
6125fveq2d 6653 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HDMap‘𝑘) = (HDMap‘𝐾))
6261, 27fveq12d 6656 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
63 hlhilset.s . . . . . . . . . . . . . . 15 𝑆 = ((HDMap‘𝐾)‘𝑊)
6462, 63eqtr4di 2854 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
6564ad2antrr 725 . . . . . . . . . . . . 13 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
6665fveq1d 6651 . . . . . . . . . . . 12 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (((HDMap‘𝑘)‘𝑤)‘𝑦) = (𝑆𝑦))
6766fveq1d 6651 . . . . . . . . . . 11 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥) = ((𝑆𝑦)‘𝑥))
6835, 35, 67mpoeq123dv 7212 . . . . . . . . . 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, 69eqtr4di 2854 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = , )
7170opeq2d 4775 . . . . . . . 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 4640 . . . . . . 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 4094 . . . . . 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 3867 . . . . 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 3867 . . . 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 3867 . . 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 499 . . 3 (𝜑𝑊𝐻)
78 tpex 7454 . . . . 5 {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∈ V
79 prex 5301 . . . . 5 {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
8078, 79unex 7453 . . . 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 6756 . 2 (𝜑 → ((HLHil‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
831, 82syl5eq 2848 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 399   = wceq 1538  wcel 2112  Vcvv 3444  csb 3831  cun 3882  {cpr 4530  {ctp 4532  cop 4534  cmpt 5113  cfv 6328  (class class class)co 7139  cmpo 7141  ndxcnx 16476   sSet csts 16477  Basecbs 16479  +gcplusg 16561  *𝑟cstv 16563  Scalarcsca 16564   ·𝑠 cvsca 16565  ·𝑖cip 16566  HLchlt 36645  LHypclh 37279  EDRingcedring 38048  DVecHcdvh 38373  HDMapchdma 39087  HGMapchg 39178  HLHilchlh 39227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-hlhil 39228
This theorem is referenced by:  hlhilsca  39230  hlhilbase  39231  hlhilplus  39232  hlhilvsca  39242  hlhilip  39243
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