Theorem lpolfN 41684

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 2734	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		eqid 2734	. . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7		eqid 2734	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolf.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 41682	. . . 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 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ⊆ wss 3899  𝒫 cpw 4552  {csn 4578  ⟶wf 6486  ‘cfv 6490  Basecbs 17134  0_gc0g 17357  LSubSpclss 20880  LSAtomsclsa 39173  LSHypclsh 39174  LPolclpoN 41679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8763  df-lpolN 41680

This theorem is referenced by: (None)