Theorem lpolpolsatN 40360

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 2733	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2733	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2733	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		lpolpolsat.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊)
7		eqid 2733	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolpolsat.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 40354	. . . 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 231	. 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 1197	. . 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 6892	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑄))
15	14	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ↔ ( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊)))
16		2fveq3 6897	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑄)))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑄 → 𝑥 = 𝑄)
18	16, 17	eqeq12d 2749	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄))
19	15, 18	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑄 → ((( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
20	19	rspcv 3609	. . . 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 486	. . 3 ⊢ ((( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄) → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)
23	12, 21, 22	syl56 36	. 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 205   ∧ wa 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949  𝒫 cpw 4603  {csn 4629  ⟶wf 6540  ‘cfv 6544  Basecbs 17144  0_gc0g 17385  LSubSpclss 20542  LSAtomsclsa 37844  LSHypclsh 37845  LPolclpoN 40351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-lpolN 40352

This theorem is referenced by: (None)