Theorem lpolsatN 42290

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 2762	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2762	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2762	. . . . 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 42285	. . . 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 18	. . 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 235	. 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 1214	. . 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 6881	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑄))
15	14	eleq1d 2847	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘𝑥) ∈ 𝐻 ↔ ( ⊥ ‘𝑄) ∈ 𝐻))
16		2fveq3 6886	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑄)))
17		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑄 → 𝑥 = 𝑄)
18	16, 17	eqeq12d 2778	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄))
19	15, 18	anbi12d 643	. . . . 5 ⊢ (𝑥 = 𝑄 → ((( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
20	19	rspcv 3576	. . . 4 ⊢ (𝑄 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) → (( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
21	13, 20	syl 18	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) → (( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)))
22		simpl 487	. . 3 ⊢ ((( ⊥ ‘𝑄) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄) → ( ⊥ ‘𝑄) ∈ 𝐻)
23	12, 21, 22	syl56 37	. 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 16	1 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝐻)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588  ⟶wf 6532  ‘cfv 6536  Basecbs 17275  0_gc0g 17498  LSubSpclss 21063  LSAtomsclsa 39776  LSHypclsh 39777  LPolclpoN 42282

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-lpolN 42283

This theorem is used by: (None)