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Theorem hlhilset 36703
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 3198 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ V)
43adantr 481 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐾 ∈ V)
52, 4syl 17 . . . 4 (𝜑𝐾 ∈ V)
6 hlhilset.h . . . . . 6 𝐻 = (LHyp‘𝐾)
7 fvex 6158 . . . . . 6 (LHyp‘𝐾) ∈ V
86, 7eqeltri 2694 . . . . 5 𝐻 ∈ V
98mptex 6440 . . . 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
10 nfcv 2761 . . . . 5 𝑘𝐾
11 nfcv 2761 . . . . . 6 𝑘𝐻
12 nfcsb1v 3530 . . . . . 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))⟩})
1311, 12nfmpt 4706 . . . . 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))⟩}))
14 fveq2 6148 . . . . . . 7 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
1514, 6syl6eqr 2673 . . . . . 6 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
16 csbeq1a 3523 . . . . . 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))⟩}))
1715, 16mpteq12dv 4693 . . . . 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))⟩})))
18 df-hlhil 36702 . . . . 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))⟩})))
1910, 13, 17, 18fvmptf 6257 . . . 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))⟩})))
205, 9, 19sylancl 693 . . 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))⟩})))
215adantr 481 . . . 4 ((𝜑𝑤 = 𝑊) → 𝐾 ∈ V)
22 fvex 6158 . . . . . 6 ((DVecH‘𝑘)‘𝑤) ∈ V
2322a1i 11 . . . . 5 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) ∈ V)
24 fvex 6158 . . . . . . 7 (Base‘𝑢) ∈ V
2524a1i 11 . . . . . 6 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) ∈ V)
26 id 22 . . . . . . . . . 10 (𝑣 = (Base‘𝑢) → 𝑣 = (Base‘𝑢))
27 id 22 . . . . . . . . . . . . 13 (𝑢 = ((DVecH‘𝑘)‘𝑤) → 𝑢 = ((DVecH‘𝑘)‘𝑤))
28 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
2928fveq2d 6152 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (DVecH‘𝑘) = (DVecH‘𝐾))
30 simplr 791 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑤 = 𝑊)
3129, 30fveq12d 6154 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
32 hlhilset.u . . . . . . . . . . . . . 14 𝑈 = ((DVecH‘𝐾)‘𝑊)
3331, 32syl6eqr 2673 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = 𝑈)
3427, 33sylan9eqr 2677 . . . . . . . . . . . 12 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → 𝑢 = 𝑈)
3534fveq2d 6152 . . . . . . . . . . 11 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = (Base‘𝑈))
36 hlhilset.v . . . . . . . . . . 11 𝑉 = (Base‘𝑈)
3735, 36syl6eqr 2673 . . . . . . . . . 10 ((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = 𝑉)
3826, 37sylan9eqr 2677 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
3938opeq2d 4377 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝑉⟩)
4034adantr 481 . . . . . . . . . . 11 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
4140fveq2d 6152 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g𝑢) = (+g𝑈))
42 hlhilset.p . . . . . . . . . 10 + = (+g𝑈)
4341, 42syl6eqr 2673 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g𝑢) = + )
4443opeq2d 4377 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(+g‘ndx), (+g𝑢)⟩ = ⟨(+g‘ndx), + ⟩)
4528fveq2d 6152 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (EDRing‘𝑘) = (EDRing‘𝐾))
4645, 30fveq12d 6154 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
47 hlhilset.e . . . . . . . . . . . . 13 𝐸 = ((EDRing‘𝐾)‘𝑊)
4846, 47syl6eqr 2673 . . . . . . . . . . . 12 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = 𝐸)
4928fveq2d 6152 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HGMap‘𝑘) = (HGMap‘𝐾))
5049, 30fveq12d 6154 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = ((HGMap‘𝐾)‘𝑊))
51 hlhilset.g . . . . . . . . . . . . . 14 𝐺 = ((HGMap‘𝐾)‘𝑊)
5250, 51syl6eqr 2673 . . . . . . . . . . . . 13 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = 𝐺)
5352opeq2d 4377 . . . . . . . . . . . 12 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩ = ⟨(*𝑟‘ndx), 𝐺⟩)
5448, 53oveq12d 6622 . . . . . . . . . . 11 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩))
55 hlhilset.r . . . . . . . . . . 11 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)
5654, 55syl6eqr 2673 . . . . . . . . . 10 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = 𝑅)
5756opeq2d 4377 . . . . . . . . 9 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5857ad2antrr 761 . . . . . . . 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⟩)
5939, 44, 58tpeq123d 4253 . . . . . . 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⟩})
6040fveq2d 6152 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
61 hlhilset.t . . . . . . . . . 10 · = ( ·𝑠𝑈)
6260, 61syl6eqr 2673 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠𝑢) = · )
6362opeq2d 4377 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
6428fveq2d 6152 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HDMap‘𝑘) = (HDMap‘𝐾))
6564, 30fveq12d 6154 . . . . . . . . . . . . . . 15 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
66 hlhilset.s . . . . . . . . . . . . . . 15 𝑆 = ((HDMap‘𝐾)‘𝑊)
6765, 66syl6eqr 2673 . . . . . . . . . . . . . 14 (((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
6867ad2antrr 761 . . . . . . . . . . . . 13 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
6968fveq1d 6150 . . . . . . . . . . . 12 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (((HDMap‘𝑘)‘𝑤)‘𝑦) = (𝑆𝑦))
7069fveq1d 6150 . . . . . . . . . . 11 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥) = ((𝑆𝑦)‘𝑥))
7138, 38, 70mpt2eq123dv 6670 . . . . . . . . . 10 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥)))
72 hlhilset.i . . . . . . . . . 10 , = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥))
7371, 72syl6eqr 2673 . . . . . . . . 9 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = , )
7473opeq2d 4377 . . . . . . . 8 (((((𝜑𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩ = ⟨(·𝑖‘ndx), , ⟩)
7563, 74preq12d 4246 . . . . . . 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), , ⟩})
7659, 75uneq12d 3746 . . . . . 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), , ⟩}))
7725, 76csbied 3541 . . . . 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), , ⟩}))
7823, 77csbied 3541 . . . 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), , ⟩}))
7921, 78csbied 3541 . . 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), , ⟩}))
802simprd 479 . . 3 (𝜑𝑊𝐻)
81 tpex 6910 . . . . 5 {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∈ V
82 prex 4870 . . . . 5 {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
8381, 82unex 6909 . . . 4 ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V
8483a1i 11 . . 3 (𝜑 → ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V)
8520, 79, 80, 84fvmptd 6245 . 2 (𝜑 → ((HLHil‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
861, 85syl5eq 2667 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 1480  wcel 1987  Vcvv 3186  csb 3514  cun 3553  {cpr 4150  {ctp 4152  cop 4154  cmpt 4673  cfv 5847  (class class class)co 6604  cmpt2 6606  ndxcnx 15778   sSet csts 15779  Basecbs 15781  +gcplusg 15862  *𝑟cstv 15864  Scalarcsca 15865   ·𝑠 cvsca 15866  ·𝑖cip 15867  HLchlt 34114  LHypclh 34747  EDRingcedring 35518  DVecHcdvh 35844  HDMapchdma 36559  HGMapchg 36652  HLHilchlh 36701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-hlhil 36702
This theorem is referenced by:  hlhilsca  36704  hlhilbase  36705  hlhilplus  36706  hlhilvsca  36716  hlhilip  36717
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