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Theorem islpolN 38097
Description: The predicate "is a polarity". (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
islpolN (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑊   𝑥, ,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem islpolN
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 lpolset.v . . . 4 𝑉 = (Base‘𝑊)
2 lpolset.s . . . 4 𝑆 = (LSubSp‘𝑊)
3 lpolset.z . . . 4 0 = (0g𝑊)
4 lpolset.a . . . 4 𝐴 = (LSAtoms‘𝑊)
5 lpolset.h . . . 4 𝐻 = (LSHyp‘𝑊)
6 lpolset.p . . . 4 𝑃 = (LPol‘𝑊)
71, 2, 3, 4, 5, 6lpolsetN 38096 . . 3 (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
87eleq2d 2846 . 2 (𝑊𝑋 → ( 𝑃 ∈ {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))}))
9 fveq1 6496 . . . . . 6 (𝑜 = → (𝑜𝑉) = ( 𝑉))
109eqeq1d 2775 . . . . 5 (𝑜 = → ((𝑜𝑉) = { 0 } ↔ ( 𝑉) = { 0 }))
11 fveq1 6496 . . . . . . . 8 (𝑜 = → (𝑜𝑦) = ( 𝑦))
12 fveq1 6496 . . . . . . . 8 (𝑜 = → (𝑜𝑥) = ( 𝑥))
1311, 12sseq12d 3885 . . . . . . 7 (𝑜 = → ((𝑜𝑦) ⊆ (𝑜𝑥) ↔ ( 𝑦) ⊆ ( 𝑥)))
1413imbi2d 333 . . . . . 6 (𝑜 = → (((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
15142albidv 1883 . . . . 5 (𝑜 = → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
1612eleq1d 2845 . . . . . . 7 (𝑜 = → ((𝑜𝑥) ∈ 𝐻 ↔ ( 𝑥) ∈ 𝐻))
17 id 22 . . . . . . . . 9 (𝑜 = 𝑜 = )
1817, 12fveq12d 6504 . . . . . . . 8 (𝑜 = → (𝑜‘(𝑜𝑥)) = ( ‘( 𝑥)))
1918eqeq1d 2775 . . . . . . 7 (𝑜 = → ((𝑜‘(𝑜𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
2016, 19anbi12d 622 . . . . . 6 (𝑜 = → (((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2120ralbidv 3142 . . . . 5 (𝑜 = → (∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2210, 15, 213anbi123d 1416 . . . 4 (𝑜 = → (((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥)) ↔ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))))
2322elrab 3590 . . 3 ( ∈ {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))} ↔ ( ∈ (𝑆𝑚 𝒫 𝑉) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))))
242fvexi 6511 . . . . 5 𝑆 ∈ V
251fvexi 6511 . . . . . 6 𝑉 ∈ V
2625pwex 5131 . . . . 5 𝒫 𝑉 ∈ V
2724, 26elmap 8234 . . . 4 ( ∈ (𝑆𝑚 𝒫 𝑉) ↔ :𝒫 𝑉𝑆)
2827anbi1i 615 . . 3 (( ∈ (𝑆𝑚 𝒫 𝑉) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))) ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))))
2923, 28bitri 267 . 2 ( ∈ {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))} ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))))
308, 29syl6bb 279 1 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069  wal 1506   = wceq 1508  wcel 2051  wral 3083  {crab 3087  wss 3824  𝒫 cpw 4417  {csn 4436  wf 6182  cfv 6186  (class class class)co 6975  𝑚 cmap 8205  Basecbs 16338  0gc0g 16568  LSubSpclss 19438  LSAtomsclsa 35588  LSHypclsh 35589  LPolclpoN 38094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-map 8207  df-lpolN 38095
This theorem is referenced by:  islpoldN  38098  lpolfN  38099  lpolvN  38100  lpolconN  38101  lpolsatN  38102  lpolpolsatN  38103
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