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Theorem islpolN 41485
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 41484 . . 3 (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
87eleq2d 2827 . 2 (𝑊𝑋 → ( 𝑃 ∈ {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))}))
9 fveq1 6905 . . . . . 6 (𝑜 = → (𝑜𝑉) = ( 𝑉))
109eqeq1d 2739 . . . . 5 (𝑜 = → ((𝑜𝑉) = { 0 } ↔ ( 𝑉) = { 0 }))
11 fveq1 6905 . . . . . . . 8 (𝑜 = → (𝑜𝑦) = ( 𝑦))
12 fveq1 6905 . . . . . . . 8 (𝑜 = → (𝑜𝑥) = ( 𝑥))
1311, 12sseq12d 4017 . . . . . . 7 (𝑜 = → ((𝑜𝑦) ⊆ (𝑜𝑥) ↔ ( 𝑦) ⊆ ( 𝑥)))
1413imbi2d 340 . . . . . 6 (𝑜 = → (((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
15142albidv 1923 . . . . 5 (𝑜 = → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
1612eleq1d 2826 . . . . . . 7 (𝑜 = → ((𝑜𝑥) ∈ 𝐻 ↔ ( 𝑥) ∈ 𝐻))
17 id 22 . . . . . . . . 9 (𝑜 = 𝑜 = )
1817, 12fveq12d 6913 . . . . . . . 8 (𝑜 = → (𝑜‘(𝑜𝑥)) = ( ‘( 𝑥)))
1918eqeq1d 2739 . . . . . . 7 (𝑜 = → ((𝑜‘(𝑜𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
2016, 19anbi12d 632 . . . . . 6 (𝑜 = → (((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2120ralbidv 3178 . . . . 5 (𝑜 = → (∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2210, 15, 213anbi123d 1438 . . . 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 3692 . . 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 6920 . . . . 5 𝑆 ∈ V
251fvexi 6920 . . . . . 6 𝑉 ∈ V
2625pwex 5380 . . . . 5 𝒫 𝑉 ∈ V
2724, 26elmap 8911 . . . 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 1087  wal 1538   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951  𝒫 cpw 4600  {csn 4626  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Basecbs 17247  0gc0g 17484  LSubSpclss 20929  LSAtomsclsa 38975  LSHypclsh 38976  LPolclpoN 41482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-lpolN 41483
This theorem is referenced by:  islpoldN  41486  lpolfN  41487  lpolvN  41488  lpolconN  41489  lpolsatN  41490  lpolpolsatN  41491
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