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Theorem lpolsatN 36596
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 2620 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2620 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2620 . . . . 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 36591 . . . 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 222 . 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 1067 . . 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 6178 . . . . . . 7 (𝑥 = 𝑄 → ( 𝑥) = ( 𝑄))
1514eleq1d 2684 . . . . . 6 (𝑥 = 𝑄 → (( 𝑥) ∈ 𝐻 ↔ ( 𝑄) ∈ 𝐻))
1614fveq2d 6182 . . . . . . 7 (𝑥 = 𝑄 → ( ‘( 𝑥)) = ( ‘( 𝑄)))
17 id 22 . . . . . . 7 (𝑥 = 𝑄𝑥 = 𝑄)
1816, 17eqeq12d 2635 . . . . . 6 (𝑥 = 𝑄 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑄)) = 𝑄))
1915, 18anbi12d 746 . . . . 5 (𝑥 = 𝑄 → ((( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (( 𝑄) ∈ 𝐻 ∧ ( ‘( 𝑄)) = 𝑄)))
2019rspcv 3300 . . . 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 473 . . 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 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  wral 2909  wss 3567  𝒫 cpw 4149  {csn 4168  wf 5872  cfv 5876  Basecbs 15838  0gc0g 16081  LSubSpclss 18913  LSAtomsclsa 34080  LSHypclsh 34081  LPolclpoN 36588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-lpolN 36589
This theorem is referenced by: (None)
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