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Theorem lpolpolsatN 41491
Description: Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolpolsat.a 𝐴 = (LSAtoms‘𝑊)
lpolpolsat.p 𝑃 = (LPol‘𝑊)
lpolpolsat.w (𝜑𝑊𝑋)
lpolpolsat.o (𝜑𝑃)
lpolpolsat.q (𝜑𝑄𝐴)
Assertion
Ref Expression
lpolpolsatN (𝜑 → ( ‘( 𝑄)) = 𝑄)

Proof of Theorem lpolpolsatN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolpolsat.o . . 3 (𝜑𝑃)
2 lpolpolsat.w . . . 4 (𝜑𝑊𝑋)
3 eqid 2737 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2737 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2737 . . . . 5 (0g𝑊) = (0g𝑊)
6 lpolpolsat.a . . . . 5 𝐴 = (LSAtoms‘𝑊)
7 eqid 2737 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolpolsat.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 41485 . . . 4 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
102, 9syl 17 . . 3 (𝜑 → ( 𝑃 ↔ ( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
111, 10mpbid 232 . 2 (𝜑 → ( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))))
12 simpr3 1197 . . 3 (( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))) → ∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))
13 lpolpolsat.q . . . 4 (𝜑𝑄𝐴)
14 fveq2 6906 . . . . . . 7 (𝑥 = 𝑄 → ( 𝑥) = ( 𝑄))
1514eleq1d 2826 . . . . . 6 (𝑥 = 𝑄 → (( 𝑥) ∈ (LSHyp‘𝑊) ↔ ( 𝑄) ∈ (LSHyp‘𝑊)))
16 2fveq3 6911 . . . . . . 7 (𝑥 = 𝑄 → ( ‘( 𝑥)) = ( ‘( 𝑄)))
17 id 22 . . . . . . 7 (𝑥 = 𝑄𝑥 = 𝑄)
1816, 17eqeq12d 2753 . . . . . 6 (𝑥 = 𝑄 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑄)) = 𝑄))
1915, 18anbi12d 632 . . . . 5 (𝑥 = 𝑄 → ((( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥) ↔ (( 𝑄) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑄)) = 𝑄)))
2019rspcv 3618 . . . 4 (𝑄𝐴 → (∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥) → (( 𝑄) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑄)) = 𝑄)))
2113, 20syl 17 . . 3 (𝜑 → (∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥) → (( 𝑄) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑄)) = 𝑄)))
22 simpr 484 . . 3 ((( 𝑄) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑄)) = 𝑄) → ( ‘( 𝑄)) = 𝑄)
2312, 21, 22syl56 36 . 2 (𝜑 → (( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))) → ( ‘( 𝑄)) = 𝑄))
2411, 23mpd 15 1 (𝜑 → ( ‘( 𝑄)) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wral 3061  wss 3951  𝒫 cpw 4600  {csn 4626  wf 6557  cfv 6561  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: (None)
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