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Theorem islpolN 39946
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 39945 . . 3 (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
87eleq2d 2823 . 2 (𝑊𝑋 → ( 𝑃 ∈ {𝑜 ∈ (𝑆m 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))}))
9 fveq1 6841 . . . . . 6 (𝑜 = → (𝑜𝑉) = ( 𝑉))
109eqeq1d 2738 . . . . 5 (𝑜 = → ((𝑜𝑉) = { 0 } ↔ ( 𝑉) = { 0 }))
11 fveq1 6841 . . . . . . . 8 (𝑜 = → (𝑜𝑦) = ( 𝑦))
12 fveq1 6841 . . . . . . . 8 (𝑜 = → (𝑜𝑥) = ( 𝑥))
1311, 12sseq12d 3977 . . . . . . 7 (𝑜 = → ((𝑜𝑦) ⊆ (𝑜𝑥) ↔ ( 𝑦) ⊆ ( 𝑥)))
1413imbi2d 340 . . . . . 6 (𝑜 = → (((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
15142albidv 1926 . . . . 5 (𝑜 = → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ↔ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥))))
1612eleq1d 2822 . . . . . . 7 (𝑜 = → ((𝑜𝑥) ∈ 𝐻 ↔ ( 𝑥) ∈ 𝐻))
17 id 22 . . . . . . . . 9 (𝑜 = 𝑜 = )
1817, 12fveq12d 6849 . . . . . . . 8 (𝑜 = → (𝑜‘(𝑜𝑥)) = ( ‘( 𝑥)))
1918eqeq1d 2738 . . . . . . 7 (𝑜 = → ((𝑜‘(𝑜𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
2016, 19anbi12d 631 . . . . . 6 (𝑜 = → (((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2120ralbidv 3174 . . . . 5 (𝑜 = → (∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥) ↔ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
2210, 15, 213anbi123d 1436 . . . 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 3645 . . 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 6856 . . . . 5 𝑆 ∈ V
251fvexi 6856 . . . . . 6 𝑉 ∈ V
2625pwex 5335 . . . . 5 𝒫 𝑉 ∈ V
2724, 26elmap 8809 . . . 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 274 . 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 286 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 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wss 3910  𝒫 cpw 4560  {csn 4586  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Basecbs 17083  0gc0g 17321  LSubSpclss 20392  LSAtomsclsa 37436  LSHypclsh 37437  LPolclpoN 39943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-lpolN 39944
This theorem is referenced by:  islpoldN  39947  lpolfN  39948  lpolvN  39949  lpolconN  39950  lpolsatN  39951  lpolpolsatN  39952
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