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Theorem lpolsetN 37257
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 3406 . 2 (𝑊𝑋𝑊 ∈ V)
2 lpolset.p . . 3 𝑃 = (LPol‘𝑊)
3 fveq2 6404 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lpolset.s . . . . . . 7 𝑆 = (LSubSp‘𝑊)
53, 4syl6eqr 2858 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6404 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lpolset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7syl6eqr 2858 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98pweqd 4356 . . . . . 6 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
105, 9oveq12d 6888 . . . . 5 (𝑤 = 𝑊 → ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) = (𝑆𝑚 𝒫 𝑉))
118fveq2d 6408 . . . . . . 7 (𝑤 = 𝑊 → (𝑜‘(Base‘𝑤)) = (𝑜𝑉))
12 fveq2 6404 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
13 lpolset.z . . . . . . . . 9 0 = (0g𝑊)
1412, 13syl6eqr 2858 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1514sneqd 4382 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1611, 15eqeq12d 2821 . . . . . 6 (𝑤 = 𝑊 → ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ↔ (𝑜𝑉) = { 0 }))
178sseq2d 3830 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑥 ⊆ (Base‘𝑤) ↔ 𝑥𝑉))
188sseq2d 3830 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑦 ⊆ (Base‘𝑤) ↔ 𝑦𝑉))
1917, 183anbi12d 1554 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) ↔ (𝑥𝑉𝑦𝑉𝑥𝑦)))
2019imbi1d 332 . . . . . . 7 (𝑤 = 𝑊 → (((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))))
21202albidv 2014 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))))
22 fveq2 6404 . . . . . . . 8 (𝑤 = 𝑊 → (LSAtoms‘𝑤) = (LSAtoms‘𝑊))
23 lpolset.a . . . . . . . 8 𝐴 = (LSAtoms‘𝑊)
2422, 23syl6eqr 2858 . . . . . . 7 (𝑤 = 𝑊 → (LSAtoms‘𝑤) = 𝐴)
25 fveq2 6404 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSHyp‘𝑤) = (LSHyp‘𝑊))
26 lpolset.h . . . . . . . . . 10 𝐻 = (LSHyp‘𝑊)
2725, 26syl6eqr 2858 . . . . . . . . 9 (𝑤 = 𝑊 → (LSHyp‘𝑤) = 𝐻)
2827eleq2d 2871 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑜𝑥) ∈ (LSHyp‘𝑤) ↔ (𝑜𝑥) ∈ 𝐻))
2928anbi1d 617 . . . . . . 7 (𝑤 = 𝑊 → (((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)))
3024, 29raleqbidv 3341 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)))
3116, 21, 303anbi123d 1553 . . . . 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 3385 . . . 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 37256 . . . 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 6902 . . . . 5 (𝑆𝑚 𝒫 𝑉) ∈ V
3534rabex 5007 . . . 4 {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))} ∈ V
3632, 33, 35fvmpt 6499 . . 3 (𝑊 ∈ V → (LPol‘𝑊) = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
372, 36syl5eq 2852 . 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 384  w3a 1100  wal 1635   = wceq 1637  wcel 2156  wral 3096  {crab 3100  Vcvv 3391  wss 3769  𝒫 cpw 4351  {csn 4370  cfv 6097  (class class class)co 6870  𝑚 cmap 8088  Basecbs 16064  0gc0g 16301  LSubSpclss 19132  LSAtomsclsa 34749  LSHypclsh 34750  LPolclpoN 37255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6060  df-fun 6099  df-fv 6105  df-ov 6873  df-lpolN 37256
This theorem is referenced by:  islpolN  37258
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