Theorem lpolfN 41509

Description: Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lpolf.v	⊢ 𝑉 = (Base‘𝑊)
lpolf.s	⊢ 𝑆 = (LSubSp‘𝑊)
lpolf.p	⊢ 𝑃 = (LPol‘𝑊)
lpolf.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)
lpolf.o	⊢ (𝜑 → ⊥ ∈ 𝑃)

Assertion

Ref	Expression
lpolfN	⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆)

Proof of Theorem lpolfN

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpolf.o	. . 3 ⊢ (𝜑 → ⊥ ∈ 𝑃)
2		lpolf.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
3		lpolf.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
4		lpolf.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
5		eqid 2736	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		eqid 2736	. . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7		eqid 2736	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolf.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 41507	. . . 4 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
10	2, 9	syl 17	. . 3 ⊢ (𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
11	1, 10	mpbid 232	. 2 ⊢ (𝜑 → ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
12	11	simpld 494	1 ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  ⟶wf 6532  ‘cfv 6536  Basecbs 17233  0_gc0g 17458  LSubSpclss 20893  LSAtomsclsa 38997  LSHypclsh 38998  LPolclpoN 41504

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-lpolN 41505

This theorem is referenced by: (None)