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Theorem lpolsetN 41465
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 3499 . 2 (𝑊𝑋𝑊 ∈ V)
2 lpolset.p . . 3 𝑃 = (LPol‘𝑊)
3 fveq2 6907 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lpolset.s . . . . . . 7 𝑆 = (LSubSp‘𝑊)
53, 4eqtr4di 2793 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lpolset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98pweqd 4622 . . . . . 6 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
105, 9oveq12d 7449 . . . . 5 (𝑤 = 𝑊 → ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) = (𝑆m 𝒫 𝑉))
118fveq2d 6911 . . . . . . 7 (𝑤 = 𝑊 → (𝑜‘(Base‘𝑤)) = (𝑜𝑉))
12 fveq2 6907 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
13 lpolset.z . . . . . . . . 9 0 = (0g𝑊)
1412, 13eqtr4di 2793 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1514sneqd 4643 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1611, 15eqeq12d 2751 . . . . . 6 (𝑤 = 𝑊 → ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ↔ (𝑜𝑉) = { 0 }))
178sseq2d 4028 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑥 ⊆ (Base‘𝑤) ↔ 𝑥𝑉))
188sseq2d 4028 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑦 ⊆ (Base‘𝑤) ↔ 𝑦𝑉))
1917, 183anbi12d 1436 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) ↔ (𝑥𝑉𝑦𝑉𝑥𝑦)))
2019imbi1d 341 . . . . . . 7 (𝑤 = 𝑊 → (((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))))
21202albidv 1921 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))))
22 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (LSAtoms‘𝑤) = (LSAtoms‘𝑊))
23 lpolset.a . . . . . . . 8 𝐴 = (LSAtoms‘𝑊)
2422, 23eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (LSAtoms‘𝑤) = 𝐴)
25 fveq2 6907 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSHyp‘𝑤) = (LSHyp‘𝑊))
26 lpolset.h . . . . . . . . . 10 𝐻 = (LSHyp‘𝑊)
2725, 26eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝑊 → (LSHyp‘𝑤) = 𝐻)
2827eleq2d 2825 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑜𝑥) ∈ (LSHyp‘𝑤) ↔ (𝑜𝑥) ∈ 𝐻))
2928anbi1d 631 . . . . . . 7 (𝑤 = 𝑊 → (((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)))
3024, 29raleqbidv 3344 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)))
3116, 21, 303anbi123d 1435 . . . . 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 3452 . . . 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 41464 . . . 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 7464 . . . . 5 (𝑆m 𝒫 𝑉) ∈ V
3534rabex 5345 . . . 4 {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))} ∈ V
3632, 33, 35fvmpt 7016 . . 3 (𝑊 ∈ V → (LPol‘𝑊) = {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
372, 36eqtrid 2787 . 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 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631  cfv 6563  (class class class)co 7431  m cmap 8865  Basecbs 17245  0gc0g 17486  LSubSpclss 20947  LSAtomsclsa 38956  LSHypclsh 38957  LPolclpoN 41463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-lpolN 41464
This theorem is referenced by:  islpolN  41466
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