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Theorem islpolN 41466
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 41465 . . 3 (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
87eleq2d 2825 . 2 (𝑊𝑋 → ( 𝑃 ∈ {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))}))
9 fveq1 6906 . . . . . 6 (𝑜 = → (𝑜𝑉) = ( 𝑉))
109eqeq1d 2737 . . . . 5 (𝑜 = → ((𝑜𝑉) = { 0 } ↔ ( 𝑉) = { 0 }))
11 fveq1 6906 . . . . . . . 8 (𝑜 = → (𝑜𝑦) = ( 𝑦))
12 fveq1 6906 . . . . . . . 8 (𝑜 = → (𝑜𝑥) = ( 𝑥))
1311, 12sseq12d 4029 . . . . . . 7 (𝑜 = → ((𝑜𝑦) ⊆ (𝑜𝑥) ↔ ( 𝑦) ⊆ ( 𝑥)))
1413imbi2d 340 . . . . . 6 (𝑜 = → (((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
15142albidv 1921 . . . . 5 (𝑜 = → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
1612eleq1d 2824 . . . . . . 7 (𝑜 = → ((𝑜𝑥) ∈ 𝐻 ↔ ( 𝑥) ∈ 𝐻))
17 id 22 . . . . . . . . 9 (𝑜 = 𝑜 = )
1817, 12fveq12d 6914 . . . . . . . 8 (𝑜 = → (𝑜‘(𝑜𝑥)) = ( ‘( 𝑥)))
1918eqeq1d 2737 . . . . . . 7 (𝑜 = → ((𝑜‘(𝑜𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
2016, 19anbi12d 632 . . . . . 6 (𝑜 = → (((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2120ralbidv 3176 . . . . 5 (𝑜 = → (∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2210, 15, 213anbi123d 1435 . . . 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 3695 . . 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 6921 . . . . 5 𝑆 ∈ V
251fvexi 6921 . . . . . 6 𝑉 ∈ V
2625pwex 5386 . . . . 5 𝒫 𝑉 ∈ V
2724, 26elmap 8910 . . . 4 ( ∈ (𝑆m 𝒫 𝑉) ↔ :𝒫 𝑉𝑆)
2827anbi1i 624 . . 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 275 . 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, 29bitrdi 287 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 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963  𝒫 cpw 4605  {csn 4631  wf 6559  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-pow 5371  ax-pr 5438  ax-un 7754
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-sbc 3792  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-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-lpolN 41464
This theorem is referenced by:  islpoldN  41467  lpolfN  41468  lpolvN  41469  lpolconN  41470  lpolsatN  41471  lpolpolsatN  41472
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