Theorem lpolpolsatN 35796

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	⊢ (𝜑 → 𝑊 ∈ 𝑋)
lpolpolsat.o	⊢ (𝜑 → ⊥ ∈ 𝑃)
lpolpolsat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)

Assertion

Ref	Expression
lpolpolsatN	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)

Proof of Theorem lpolpolsatN

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

Step	Hyp	Ref	Expression
1		lpolpolsat.o	. . 3 ⊢ (𝜑 → ⊥ ∈ 𝑃)
2		lpolpolsat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
3		eqid 2610	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2610	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2610	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		lpolpolsat.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊)
7		eqid 2610	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolpolsat.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 35790	. . . 4 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘(Base‘𝑊)) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
10	2, 9	syl 17	. . 3 ⊢ (𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘(Base‘𝑊)) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
11	1, 10	mpbid 221	. 2 ⊢ (𝜑 → ( ⊥ :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘(Base‘𝑊)) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
12		simpr3 1062	. . 3 ⊢ (( ⊥ :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘(Base‘𝑊)) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
13		lpolpolsat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
14		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑄))
15	14	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ↔ ( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊)))
16	14	fveq2d 6107	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑄)))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑄 → 𝑥 = 𝑄)
18	16, 17	eqeq12d 2625	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄))
19	15, 18	anbi12d 743	. . . . 5 ⊢ (𝑥 = 𝑄 → ((( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
20	19	rspcv 3278	. . . 4 ⊢ (𝑄 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) → (( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
21	13, 20	syl 17	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) → (( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
22		simpr 476	. . 3 ⊢ ((( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄) → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)
23	12, 21, 22	syl56 35	. 2 ⊢ (𝜑 → (( ⊥ :𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘(Base‘𝑊)) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑊) ∧ 𝑦 ⊆ (Base‘𝑊) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄))
24	11, 23	mpd 15	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  0_gc0g 15923  LSubSpclss 18753  LSAtomsclsa 33279  LSHypclsh 33280  LPolclpoN 35787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lpolN 35788

This theorem is referenced by: (None)