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Theorem lpolsatN 40853
Description: The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolsat.a 𝐴 = (LSAtoms‘𝑊)
lpolsat.h 𝐻 = (LSHyp‘𝑊)
lpolsat.p 𝑃 = (LPol‘𝑊)
lpolsat.w (𝜑𝑊𝑋)
lpolsat.o (𝜑𝑃)
lpolsat.q (𝜑𝑄𝐴)
Assertion
Ref Expression
lpolsatN (𝜑 → ( 𝑄) ∈ 𝐻)

Proof of Theorem lpolsatN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolsat.o . . 3 (𝜑𝑃)
2 lpolsat.w . . . 4 (𝜑𝑊𝑋)
3 eqid 2724 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2724 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2724 . . . . 5 (0g𝑊) = (0g𝑊)
6 lpolsat.a . . . . 5 𝐴 = (LSAtoms‘𝑊)
7 lpolsat.h . . . . 5 𝐻 = (LSHyp‘𝑊)
8 lpolsat.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 40848 . . . 4 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 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) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))))
111, 10mpbid 231 . 2 (𝜑 → ( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))))
12 simpr3 1193 . . 3 (( :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ‘(Base‘𝑊)) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))) → ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))
13 lpolsat.q . . . 4 (𝜑𝑄𝐴)
14 fveq2 6882 . . . . . . 7 (𝑥 = 𝑄 → ( 𝑥) = ( 𝑄))
1514eleq1d 2810 . . . . . 6 (𝑥 = 𝑄 → (( 𝑥) ∈ 𝐻 ↔ ( 𝑄) ∈ 𝐻))
16 2fveq3 6887 . . . . . . 7 (𝑥 = 𝑄 → ( ‘( 𝑥)) = ( ‘( 𝑄)))
17 id 22 . . . . . . 7 (𝑥 = 𝑄𝑥 = 𝑄)
1816, 17eqeq12d 2740 . . . . . 6 (𝑥 = 𝑄 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑄)) = 𝑄))
1915, 18anbi12d 630 . . . . 5 (𝑥 = 𝑄 → ((( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (( 𝑄) ∈ 𝐻 ∧ ( ‘( 𝑄)) = 𝑄)))
2019rspcv 3600 . . . 4 (𝑄𝐴 → (∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥) → (( 𝑄) ∈ 𝐻 ∧ ( ‘( 𝑄)) = 𝑄)))
2113, 20syl 17 . . 3 (𝜑 → (∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥) → (( 𝑄) ∈ 𝐻 ∧ ( ‘( 𝑄)) = 𝑄)))
22 simpl 482 . . 3 ((( 𝑄) ∈ 𝐻 ∧ ( ‘( 𝑄)) = 𝑄) → ( 𝑄) ∈ 𝐻)
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) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))) → ( 𝑄) ∈ 𝐻))
2411, 23mpd 15 1 (𝜑 → ( 𝑄) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3053  wss 3941  𝒫 cpw 4595  {csn 4621  wf 6530  cfv 6534  Basecbs 17145  0gc0g 17386  LSubSpclss 20770  LSAtomsclsa 38338  LSHypclsh 38339  LPolclpoN 40845
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-lpolN 40846
This theorem is referenced by: (None)
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