Theorem lpolpolsatN 38640

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 2821	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2821	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2821	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		lpolpolsat.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊)
7		eqid 2821	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolpolsat.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 38634	. . . 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 234	. 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 1192	. . 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 6670	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑄))
15	14	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = 𝑄 → (( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ↔ ( ⊥ ‘𝑄) ∈ (LSHyp‘𝑊)))
16		2fveq3 6675	. . . . . . 7 ⊢ (𝑥 = 𝑄 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑄)))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑄 → 𝑥 = 𝑄)
18	16, 17	eqeq12d 2837	. . . . . 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 3618	. . . 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 487	. . 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 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567  ⟶wf 6351  ‘cfv 6355  Basecbs 16483  0_gc0g 16713  LSubSpclss 19703  LSAtomsclsa 36125  LSHypclsh 36126  LPolclpoN 38631

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-lpolN 38632

This theorem is referenced by: (None)