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Theorem lpolpolsatN 39240
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 39234 . . . 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 235 . 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 1198 . . 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 6717 . . . . . . 7 (𝑥 = 𝑄 → ( 𝑥) = ( 𝑄))
1514eleq1d 2822 . . . . . 6 (𝑥 = 𝑄 → (( 𝑥) ∈ (LSHyp‘𝑊) ↔ ( 𝑄) ∈ (LSHyp‘𝑊)))
16 2fveq3 6722 . . . . . . 7 (𝑥 = 𝑄 → ( ‘( 𝑥)) = ( ‘( 𝑄)))
17 id 22 . . . . . . 7 (𝑥 = 𝑄𝑥 = 𝑄)
1816, 17eqeq12d 2753 . . . . . 6 (𝑥 = 𝑄 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑄)) = 𝑄))
1915, 18anbi12d 634 . . . . 5 (𝑥 = 𝑄 → ((( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥) ↔ (( 𝑄) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑄)) = 𝑄)))
2019rspcv 3532 . . . 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 488 . . 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 209  wa 399  w3a 1089  wal 1541   = wceq 1543  wcel 2110  wral 3061  wss 3866  𝒫 cpw 4513  {csn 4541  wf 6376  cfv 6380  Basecbs 16760  0gc0g 16944  LSubSpclss 19968  LSAtomsclsa 36725  LSHypclsh 36726  LPolclpoN 39231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-lpolN 39232
This theorem is referenced by: (None)
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