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Theorem islpolN 39996
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 39995 . . 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 2820 . 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 6845 . . . . . 6 (π‘œ = βŠ₯ β†’ (π‘œβ€˜π‘‰) = ( βŠ₯ β€˜π‘‰))
109eqeq1d 2735 . . . . 5 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜π‘‰) = { 0 } ↔ ( βŠ₯ β€˜π‘‰) = { 0 }))
11 fveq1 6845 . . . . . . . 8 (π‘œ = βŠ₯ β†’ (π‘œβ€˜π‘¦) = ( βŠ₯ β€˜π‘¦))
12 fveq1 6845 . . . . . . . 8 (π‘œ = βŠ₯ β†’ (π‘œβ€˜π‘₯) = ( βŠ₯ β€˜π‘₯))
1311, 12sseq12d 3981 . . . . . . 7 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯) ↔ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)))
1413imbi2d 341 . . . . . 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 1927 . . . . 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 2819 . . . . . . 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 6853 . . . . . . . 8 (π‘œ = βŠ₯ β†’ (π‘œβ€˜(π‘œβ€˜π‘₯)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)))
1918eqeq1d 2735 . . . . . . 7 (π‘œ = βŠ₯ β†’ ((π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))
2016, 19anbi12d 632 . . . . . 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 3171 . . . . 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 1437 . . . 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 3649 . . 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 6860 . . . . 5 𝑆 ∈ V
251fvexi 6860 . . . . . 6 𝑉 ∈ V
2625pwex 5339 . . . . 5 𝒫 𝑉 ∈ V
2724, 26elmap 8815 . . . 4 ( βŠ₯ ∈ (𝑆 ↑m 𝒫 𝑉) ↔ βŠ₯ :𝒫 π‘‰βŸΆπ‘†)
2827anbi1i 625 . . 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 275 . 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 287 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 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   βŠ† wss 3914  π’« cpw 4564  {csn 4590  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  Basecbs 17091  0gc0g 17329  LSubSpclss 20436  LSAtomsclsa 37486  LSHypclsh 37487  LPolclpoN 39993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-lpolN 39994
This theorem is referenced by:  islpoldN  39997  lpolfN  39998  lpolvN  39999  lpolconN  40000  lpolsatN  40001  lpolpolsatN  40002
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