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Theorem lpolsetN 40866
Description: The set of polarities of a left module or left vector space. (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
lpolsetN (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‘₯))})
Distinct variable groups:   π‘₯,𝐴   𝑆,π‘œ   π‘œ,𝑉   π‘₯,π‘œ,𝑦,π‘Š
Allowed substitution hints:   𝐴(𝑦,π‘œ)   𝑃(π‘₯,𝑦,π‘œ)   𝑆(π‘₯,𝑦)   𝐻(π‘₯,𝑦,π‘œ)   𝑉(π‘₯,𝑦)   𝑋(π‘₯,𝑦,π‘œ)   0 (π‘₯,𝑦,π‘œ)

Proof of Theorem lpolsetN
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 elex 3487 . 2 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
2 lpolset.p . . 3 𝑃 = (LPolβ€˜π‘Š)
3 fveq2 6885 . . . . . . 7 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
4 lpolset.s . . . . . . 7 𝑆 = (LSubSpβ€˜π‘Š)
53, 4eqtr4di 2784 . . . . . 6 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝑆)
6 fveq2 6885 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
7 lpolset.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
86, 7eqtr4di 2784 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
98pweqd 4614 . . . . . 6 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
105, 9oveq12d 7423 . . . . 5 (𝑀 = π‘Š β†’ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) = (𝑆 ↑m 𝒫 𝑉))
118fveq2d 6889 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘œβ€˜(Baseβ€˜π‘€)) = (π‘œβ€˜π‘‰))
12 fveq2 6885 . . . . . . . . 9 (𝑀 = π‘Š β†’ (0gβ€˜π‘€) = (0gβ€˜π‘Š))
13 lpolset.z . . . . . . . . 9 0 = (0gβ€˜π‘Š)
1412, 13eqtr4di 2784 . . . . . . . 8 (𝑀 = π‘Š β†’ (0gβ€˜π‘€) = 0 )
1514sneqd 4635 . . . . . . 7 (𝑀 = π‘Š β†’ {(0gβ€˜π‘€)} = { 0 })
1611, 15eqeq12d 2742 . . . . . 6 (𝑀 = π‘Š β†’ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ↔ (π‘œβ€˜π‘‰) = { 0 }))
178sseq2d 4009 . . . . . . . . 9 (𝑀 = π‘Š β†’ (π‘₯ βŠ† (Baseβ€˜π‘€) ↔ π‘₯ βŠ† 𝑉))
188sseq2d 4009 . . . . . . . . 9 (𝑀 = π‘Š β†’ (𝑦 βŠ† (Baseβ€˜π‘€) ↔ 𝑦 βŠ† 𝑉))
1917, 183anbi12d 1433 . . . . . . . 8 (𝑀 = π‘Š β†’ ((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) ↔ (π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦)))
2019imbi1d 341 . . . . . . 7 (𝑀 = π‘Š β†’ (((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ↔ ((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯))))
21202albidv 1918 . . . . . 6 (𝑀 = π‘Š β†’ (βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ 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†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯))))
22 fveq2 6885 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSAtomsβ€˜π‘€) = (LSAtomsβ€˜π‘Š))
23 lpolset.a . . . . . . . 8 𝐴 = (LSAtomsβ€˜π‘Š)
2422, 23eqtr4di 2784 . . . . . . 7 (𝑀 = π‘Š β†’ (LSAtomsβ€˜π‘€) = 𝐴)
25 fveq2 6885 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSHypβ€˜π‘€) = (LSHypβ€˜π‘Š))
26 lpolset.h . . . . . . . . . 10 𝐻 = (LSHypβ€˜π‘Š)
2725, 26eqtr4di 2784 . . . . . . . . 9 (𝑀 = π‘Š β†’ (LSHypβ€˜π‘€) = 𝐻)
2827eleq2d 2813 . . . . . . . 8 (𝑀 = π‘Š β†’ ((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ↔ (π‘œβ€˜π‘₯) ∈ 𝐻))
2928anbi1d 629 . . . . . . 7 (𝑀 = π‘Š β†’ (((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯) ↔ ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯)))
3024, 29raleqbidv 3336 . . . . . 6 (𝑀 = π‘Š β†’ (βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝐴 ((π‘œβ€˜π‘₯) ∈ 𝐻 ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯)))
3116, 21, 303anbi123d 1432 . . . . 5 (𝑀 = π‘Š β†’ (((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯)) ↔ ((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‘₯))))
3210, 31rabeqbidv 3443 . . . 4 (𝑀 = π‘Š β†’ {π‘œ ∈ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) ∣ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))} = {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‘₯))})
33 df-lpolN 40865 . . . 4 LPol = (𝑀 ∈ V ↦ {π‘œ ∈ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) ∣ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))})
34 ovex 7438 . . . . 5 (𝑆 ↑m 𝒫 𝑉) ∈ V
3534rabex 5325 . . . 4 {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‘₯))} ∈ V
3632, 33, 35fvmpt 6992 . . 3 (π‘Š ∈ V β†’ (LPolβ€˜π‘Š) = {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‘₯))})
372, 36eqtrid 2778 . 2 (π‘Š ∈ V 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‘₯))})
381, 37syl 17 1 (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‘₯))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597  {csn 4623  β€˜cfv 6537  (class class class)co 7405   ↑m cmap 8822  Basecbs 17153  0gc0g 17394  LSubSpclss 20778  LSAtomsclsa 38357  LSHypclsh 38358  LPolclpoN 40864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-lpolN 40865
This theorem is referenced by:  islpolN  40867
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