Theorem islpolN 39497

Description: The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lpolset.v	⊢ 𝑉 = (Base‘𝑊)
lpolset.s	⊢ 𝑆 = (LSubSp‘𝑊)
lpolset.z	⊢ 0 = (0g‘𝑊)
lpolset.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lpolset.h	⊢ 𝐻 = (LSHyp‘𝑊)
lpolset.p	⊢ 𝑃 = (LPol‘𝑊)

Assertion

Ref	Expression
islpolN	⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑊   𝑥, ⊥ ,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem islpolN

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpolset.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		lpolset.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lpolset.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		lpolset.a	. . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lpolset.h	. . . 4 ⊢ 𝐻 = (LSHyp‘𝑊)
6		lpolset.p	. . . 4 ⊢ 𝑃 = (LPol‘𝑊)
7	1, 2, 3, 4, 5, 6	lpolsetN 39496	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑃 = {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
8	7	eleq2d 2824	. 2 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ⊥ ∈ {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))}))
9		fveq1 6773	. . . . . 6 ⊢ (𝑜 = ⊥ → (𝑜‘𝑉) = ( ⊥ ‘𝑉))
10	9	eqeq1d 2740	. . . . 5 ⊢ (𝑜 = ⊥ → ((𝑜‘𝑉) = { 0 } ↔ ( ⊥ ‘𝑉) = { 0 }))
11		fveq1 6773	. . . . . . . 8 ⊢ (𝑜 = ⊥ → (𝑜‘𝑦) = ( ⊥ ‘𝑦))
12		fveq1 6773	. . . . . . . 8 ⊢ (𝑜 = ⊥ → (𝑜‘𝑥) = ( ⊥ ‘𝑥))
13	11, 12	sseq12d 3954	. . . . . . 7 ⊢ (𝑜 = ⊥ → ((𝑜‘𝑦) ⊆ (𝑜‘𝑥) ↔ ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
14	13	imbi2d 341	. . . . . 6 ⊢ (𝑜 = ⊥ → (((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ↔ ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))))
15	14	2albidv 1926	. . . . 5 ⊢ (𝑜 = ⊥ → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ↔ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))))
16	12	eleq1d 2823	. . . . . . 7 ⊢ (𝑜 = ⊥ → ((𝑜‘𝑥) ∈ 𝐻 ↔ ( ⊥ ‘𝑥) ∈ 𝐻))
17		id 22	. . . . . . . . 9 ⊢ (𝑜 = ⊥ → 𝑜 = ⊥ )
18	17, 12	fveq12d 6781	. . . . . . . 8 ⊢ (𝑜 = ⊥ → (𝑜‘(𝑜‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑥)))
19	18	eqeq1d 2740	. . . . . . 7 ⊢ (𝑜 = ⊥ → ((𝑜‘(𝑜‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
20	16, 19	anbi12d 631	. . . . . 6 ⊢ (𝑜 = ⊥ → (((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))
21	20	ralbidv 3112	. . . . 5 ⊢ (𝑜 = ⊥ → (∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))
22	10, 15, 21	3anbi123d 1435	. . . 4 ⊢ (𝑜 = ⊥ → (((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)) ↔ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
23	22	elrab 3624	. . 3 ⊢ ( ⊥ ∈ {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))} ↔ ( ⊥ ∈ (𝑆 ↑m 𝒫 𝑉) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
24	2	fvexi 6788	. . . . 5 ⊢ 𝑆 ∈ V
25	1	fvexi 6788	. . . . . 6 ⊢ 𝑉 ∈ V
26	25	pwex 5303	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
27	24, 26	elmap 8659	. . . 4 ⊢ ( ⊥ ∈ (𝑆 ↑m 𝒫 𝑉) ↔ ⊥ :𝒫 𝑉⟶𝑆)
28	27	anbi1i 624	. . 3 ⊢ (( ⊥ ∈ (𝑆 ↑m 𝒫 𝑉) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
29	23, 28	bitri 274	. 2 ⊢ ( ⊥ ∈ {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))} ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
30	8, 29	bitrdi 287	1 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Basecbs 16912  0_gc0g 17150  LSubSpclss 20193  LSAtomsclsa 36988  LSHypclsh 36989  LPolclpoN 39494

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-lpolN 39495

This theorem is referenced by:  islpoldN  39498  lpolfN  39499  lpolvN  39500  lpolconN  39501  lpolsatN  39502  lpolpolsatN  39503