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Theorem lpolpolsatN 36297
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 2621 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2621 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2621 . . . . 5 (0g𝑊) = (0g𝑊)
6 lpolpolsat.a . . . . 5 𝐴 = (LSAtoms‘𝑊)
7 eqid 2621 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolpolsat.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 36291 . . . 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 222 . 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 1067 . . 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 6158 . . . . . . 7 (𝑥 = 𝑄 → ( 𝑥) = ( 𝑄))
1514eleq1d 2683 . . . . . 6 (𝑥 = 𝑄 → (( 𝑥) ∈ (LSHyp‘𝑊) ↔ ( 𝑄) ∈ (LSHyp‘𝑊)))
1614fveq2d 6162 . . . . . . 7 (𝑥 = 𝑄 → ( ‘( 𝑥)) = ( ‘( 𝑄)))
17 id 22 . . . . . . 7 (𝑥 = 𝑄𝑥 = 𝑄)
1816, 17eqeq12d 2636 . . . . . 6 (𝑥 = 𝑄 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑄)) = 𝑄))
1915, 18anbi12d 746 . . . . 5 (𝑥 = 𝑄 → ((( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥) ↔ (( 𝑄) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑄)) = 𝑄)))
2019rspcv 3295 . . . 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 477 . . 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 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2908  wss 3560  𝒫 cpw 4136  {csn 4155  wf 5853  cfv 5857  Basecbs 15800  0gc0g 16040  LSubSpclss 18872  LSAtomsclsa 33780  LSHypclsh 33781  LPolclpoN 36288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-lpolN 36289
This theorem is referenced by: (None)
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