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Theorem islpolN 35790
 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 35789 . . 3 (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
87eleq2d 2673 . 2 (𝑊𝑋 → ( 𝑃 ∈ {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))}))
9 fveq1 6102 . . . . . 6 (𝑜 = → (𝑜𝑉) = ( 𝑉))
109eqeq1d 2612 . . . . 5 (𝑜 = → ((𝑜𝑉) = { 0 } ↔ ( 𝑉) = { 0 }))
11 fveq1 6102 . . . . . . . 8 (𝑜 = → (𝑜𝑦) = ( 𝑦))
12 fveq1 6102 . . . . . . . 8 (𝑜 = → (𝑜𝑥) = ( 𝑥))
1311, 12sseq12d 3597 . . . . . . 7 (𝑜 = → ((𝑜𝑦) ⊆ (𝑜𝑥) ↔ ( 𝑦) ⊆ ( 𝑥)))
1413imbi2d 329 . . . . . 6 (𝑜 = → (((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
15142albidv 1838 . . . . 5 (𝑜 = → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
1612eleq1d 2672 . . . . . . 7 (𝑜 = → ((𝑜𝑥) ∈ 𝐻 ↔ ( 𝑥) ∈ 𝐻))
17 id 22 . . . . . . . . 9 (𝑜 = 𝑜 = )
1817, 12fveq12d 6109 . . . . . . . 8 (𝑜 = → (𝑜‘(𝑜𝑥)) = ( ‘( 𝑥)))
1918eqeq1d 2612 . . . . . . 7 (𝑜 = → ((𝑜‘(𝑜𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
2016, 19anbi12d 743 . . . . . 6 (𝑜 = → (((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2120ralbidv 2969 . . . . 5 (𝑜 = → (∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2210, 15, 213anbi123d 1391 . . . 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 3331 . . 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))))
24 fvex 6113 . . . . . 6 (LSubSp‘𝑊) ∈ V
252, 24eqeltri 2684 . . . . 5 𝑆 ∈ V
26 fvex 6113 . . . . . . 7 (Base‘𝑊) ∈ V
271, 26eqeltri 2684 . . . . . 6 𝑉 ∈ V
2827pwex 4774 . . . . 5 𝒫 𝑉 ∈ V
2925, 28elmap 7772 . . . 4 ( ∈ (𝑆𝑚 𝒫 𝑉) ↔ :𝒫 𝑉𝑆)
3029anbi1i 727 . . 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))))
3123, 30bitri 263 . 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))))
328, 31syl6bb 275 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 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Basecbs 15695  0gc0g 15923  LSubSpclss 18753  LSAtomsclsa 33279  LSHypclsh 33280  LPolclpoN 35787 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lpolN 35788 This theorem is referenced by:  islpoldN  35791  lpolfN  35792  lpolvN  35793  lpolconN  35794  lpolsatN  35795  lpolpolsatN  35796
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