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Theorem islpolN 40349
Description: The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolset.v 𝑉 = (Baseβ€˜π‘Š)
lpolset.s 𝑆 = (LSubSpβ€˜π‘Š)
lpolset.z 0 = (0gβ€˜π‘Š)
lpolset.a 𝐴 = (LSAtomsβ€˜π‘Š)
lpolset.h 𝐻 = (LSHypβ€˜π‘Š)
lpolset.p 𝑃 = (LPolβ€˜π‘Š)
Assertion
Ref Expression
islpolN (π‘Š ∈ 𝑋 β†’ ( βŠ₯ ∈ 𝑃 ↔ ( βŠ₯ :𝒫 π‘‰βŸΆπ‘† ∧ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑦,π‘Š   π‘₯, βŠ₯ ,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝑃(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   𝐻(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   𝑋(π‘₯,𝑦)   0 (π‘₯,𝑦)

Proof of Theorem islpolN
Dummy variable π‘œ is distinct from all other variables.
StepHypRef Expression
1 lpolset.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
2 lpolset.s . . . 4 𝑆 = (LSubSpβ€˜π‘Š)
3 lpolset.z . . . 4 0 = (0gβ€˜π‘Š)
4 lpolset.a . . . 4 𝐴 = (LSAtomsβ€˜π‘Š)
5 lpolset.h . . . 4 𝐻 = (LSHypβ€˜π‘Š)
6 lpolset.p . . . 4 𝑃 = (LPolβ€˜π‘Š)
71, 2, 3, 4, 5, 6lpolsetN 40348 . . 3 (π‘Š ∈ 𝑋 β†’ 𝑃 = {π‘œ ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((π‘œβ€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))})
87eleq2d 2819 . 2 (π‘Š ∈ 𝑋 β†’ ( βŠ₯ ∈ 𝑃 ↔ βŠ₯ ∈ {π‘œ ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((π‘œβ€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))}))
9 fveq1 6890 . . . . . 6 (π‘œ = βŠ₯ β†’ (π‘œβ€˜π‘‰) = ( βŠ₯ β€˜π‘‰))
109eqeq1d 2734 . . . . 5 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜π‘‰) = { 0 } ↔ ( βŠ₯ β€˜π‘‰) = { 0 }))
11 fveq1 6890 . . . . . . . 8 (π‘œ = βŠ₯ β†’ (π‘œβ€˜π‘¦) = ( βŠ₯ β€˜π‘¦))
12 fveq1 6890 . . . . . . . 8 (π‘œ = βŠ₯ β†’ (π‘œβ€˜π‘₯) = ( βŠ₯ β€˜π‘₯))
1311, 12sseq12d 4015 . . . . . . 7 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯) ↔ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)))
1413imbi2d 340 . . . . . 6 (π‘œ = βŠ₯ β†’ (((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ↔ ((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯))))
15142albidv 1926 . . . . 5 (π‘œ = βŠ₯ β†’ (βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ↔ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯))))
1612eleq1d 2818 . . . . . . 7 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜π‘₯) ∈ 𝐻 ↔ ( βŠ₯ β€˜π‘₯) ∈ 𝐻))
17 id 22 . . . . . . . . 9 (π‘œ = βŠ₯ β†’ π‘œ = βŠ₯ )
1817, 12fveq12d 6898 . . . . . . . 8 (π‘œ = βŠ₯ β†’ (π‘œβ€˜(π‘œβ€˜π‘₯)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)))
1918eqeq1d 2734 . . . . . . 7 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))
2016, 19anbi12d 631 . . . . . 6 (π‘œ = βŠ₯ β†’ (((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯) ↔ (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))
2120ralbidv 3177 . . . . 5 (π‘œ = βŠ₯ β†’ (βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))
2210, 15, 213anbi123d 1436 . . . 4 (π‘œ = βŠ₯ β†’ (((π‘œβ€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯)) ↔ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
2322elrab 3683 . . 3 ( βŠ₯ ∈ {π‘œ ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((π‘œβ€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))} ↔ ( βŠ₯ ∈ (𝑆 ↑m 𝒫 𝑉) ∧ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
242fvexi 6905 . . . . 5 𝑆 ∈ V
251fvexi 6905 . . . . . 6 𝑉 ∈ V
2625pwex 5378 . . . . 5 𝒫 𝑉 ∈ V
2724, 26elmap 8864 . . . 4 ( βŠ₯ ∈ (𝑆 ↑m 𝒫 𝑉) ↔ βŠ₯ :𝒫 π‘‰βŸΆπ‘†)
2827anbi1i 624 . . 3 (( βŠ₯ ∈ (𝑆 ↑m 𝒫 𝑉) ∧ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) ↔ ( βŠ₯ :𝒫 π‘‰βŸΆπ‘† ∧ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
2923, 28bitri 274 . 2 ( βŠ₯ ∈ {π‘œ ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((π‘œβ€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))} ↔ ( βŠ₯ :𝒫 π‘‰βŸΆπ‘† ∧ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
308, 29bitrdi 286 1 (π‘Š ∈ 𝑋 β†’ ( βŠ₯ ∈ 𝑃 ↔ ( βŠ₯ :𝒫 π‘‰βŸΆπ‘† ∧ (( βŠ₯ β€˜π‘‰) = { 0 } ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  Basecbs 17143  0gc0g 17384  LSubSpclss 20541  LSAtomsclsa 37839  LSHypclsh 37840  LPolclpoN 40346
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-lpolN 40347
This theorem is referenced by:  islpoldN  40350  lpolfN  40351  lpolvN  40352  lpolconN  40353  lpolsatN  40354  lpolpolsatN  40355
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