Theorem lpolsatN 40853

Description: The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lpolsat.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lpolsat.h	⊢ 𝐻 = (LSHyp‘𝑊)
lpolsat.p	⊢ 𝑃 = (LPol‘𝑊)
lpolsat.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)
lpolsat.o	⊢ (𝜑 → ⊥ ∈ 𝑃)
lpolsat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)

Assertion

Ref	Expression
lpolsatN	⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝐻)

Proof of Theorem lpolsatN

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

Step	Hyp	Ref	Expression
1		lpolsat.o	. . 3 ⊢ (𝜑 → ⊥ ∈ 𝑃)
2		lpolsat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
3		eqid 2724	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2724	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2724	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		lpolsat.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊)
7		lpolsat.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
8		lpolsat.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 40848	. . . 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) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘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) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘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) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
12		simpr3 1193	. . 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) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
13		lpolsat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
14		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑄))
15	14	eleq1d 2810	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘𝑥) ∈ 𝐻 ↔ ( ⊥ ‘𝑄) ∈ 𝐻))
16		2fveq3 6887	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑄)))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑄 → 𝑥 = 𝑄)
18	16, 17	eqeq12d 2740	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄))
19	15, 18	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝑄 → ((( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
20	19	rspcv 3600	. . . 4 ⊢ (𝑄 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) → (( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
21	13, 20	syl 17	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) → (( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
22		simpl 482	. . 3 ⊢ ((( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄) → ( ⊥ ‘𝑄) ∈ 𝐻)
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) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ( ⊥ ‘𝑄) ∈ 𝐻))
24	11, 23	mpd 15	1 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝐻)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∀wral 3053   ⊆ wss 3941  𝒫 cpw 4595  {csn 4621  ⟶wf 6530  ‘cfv 6534  Basecbs 17145  0_gc0g 17386  LSubSpclss 20770  LSAtomsclsa 38338  LSHypclsh 38339  LPolclpoN 40845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-lpolN 40846

This theorem is referenced by: (None)