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Theorem lpolsetN 37556
Description: The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolset.v 𝑉 = (Base‘𝑊)
lpolset.s 𝑆 = (LSubSp‘𝑊)
lpolset.z 0 = (0g𝑊)
lpolset.a 𝐴 = (LSAtoms‘𝑊)
lpolset.h 𝐻 = (LSHyp‘𝑊)
lpolset.p 𝑃 = (LPol‘𝑊)
Assertion
Ref Expression
lpolsetN (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
Distinct variable groups:   𝑥,𝐴   𝑆,𝑜   𝑜,𝑉   𝑥,𝑜,𝑦,𝑊
Allowed substitution hints:   𝐴(𝑦,𝑜)   𝑃(𝑥,𝑦,𝑜)   𝑆(𝑥,𝑦)   𝐻(𝑥,𝑦,𝑜)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑦,𝑜)   0 (𝑥,𝑦,𝑜)

Proof of Theorem lpolsetN
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3428 . 2 (𝑊𝑋𝑊 ∈ V)
2 lpolset.p . . 3 𝑃 = (LPol‘𝑊)
3 fveq2 6432 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lpolset.s . . . . . . 7 𝑆 = (LSubSp‘𝑊)
53, 4syl6eqr 2878 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6432 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lpolset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7syl6eqr 2878 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98pweqd 4382 . . . . . 6 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
105, 9oveq12d 6922 . . . . 5 (𝑤 = 𝑊 → ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) = (𝑆𝑚 𝒫 𝑉))
118fveq2d 6436 . . . . . . 7 (𝑤 = 𝑊 → (𝑜‘(Base‘𝑤)) = (𝑜𝑉))
12 fveq2 6432 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
13 lpolset.z . . . . . . . . 9 0 = (0g𝑊)
1412, 13syl6eqr 2878 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1514sneqd 4408 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1611, 15eqeq12d 2839 . . . . . 6 (𝑤 = 𝑊 → ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ↔ (𝑜𝑉) = { 0 }))
178sseq2d 3857 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑥 ⊆ (Base‘𝑤) ↔ 𝑥𝑉))
188sseq2d 3857 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑦 ⊆ (Base‘𝑤) ↔ 𝑦𝑉))
1917, 183anbi12d 1567 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) ↔ (𝑥𝑉𝑦𝑉𝑥𝑦)))
2019imbi1d 333 . . . . . . 7 (𝑤 = 𝑊 → (((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))))
21202albidv 2024 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))))
22 fveq2 6432 . . . . . . . 8 (𝑤 = 𝑊 → (LSAtoms‘𝑤) = (LSAtoms‘𝑊))
23 lpolset.a . . . . . . . 8 𝐴 = (LSAtoms‘𝑊)
2422, 23syl6eqr 2878 . . . . . . 7 (𝑤 = 𝑊 → (LSAtoms‘𝑤) = 𝐴)
25 fveq2 6432 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSHyp‘𝑤) = (LSHyp‘𝑊))
26 lpolset.h . . . . . . . . . 10 𝐻 = (LSHyp‘𝑊)
2725, 26syl6eqr 2878 . . . . . . . . 9 (𝑤 = 𝑊 → (LSHyp‘𝑤) = 𝐻)
2827eleq2d 2891 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑜𝑥) ∈ (LSHyp‘𝑤) ↔ (𝑜𝑥) ∈ 𝐻))
2928anbi1d 625 . . . . . . 7 (𝑤 = 𝑊 → (((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)))
3024, 29raleqbidv 3363 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)))
3116, 21, 303anbi123d 1566 . . . . 5 (𝑤 = 𝑊 → (((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)) ↔ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))))
3210, 31rabeqbidv 3407 . . . 4 (𝑤 = 𝑊 → {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))} = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
33 df-lpolN 37555 . . . 4 LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
34 ovex 6936 . . . . 5 (𝑆𝑚 𝒫 𝑉) ∈ V
3534rabex 5036 . . . 4 {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))} ∈ V
3632, 33, 35fvmpt 6528 . . 3 (𝑊 ∈ V → (LPol‘𝑊) = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
372, 36syl5eq 2872 . 2 (𝑊 ∈ V → 𝑃 = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
381, 37syl 17 1 (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113  wal 1656   = wceq 1658  wcel 2166  wral 3116  {crab 3120  Vcvv 3413  wss 3797  𝒫 cpw 4377  {csn 4396  cfv 6122  (class class class)co 6904  𝑚 cmap 8121  Basecbs 16221  0gc0g 16452  LSubSpclss 19287  LSAtomsclsa 35048  LSHypclsh 35049  LPolclpoN 37554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-iota 6085  df-fun 6124  df-fv 6130  df-ov 6907  df-lpolN 37555
This theorem is referenced by:  islpolN  37557
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