Theorem lpolsetN 42314

Description: The set of polarities of a left module or left vector space. (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
lpolsetN	⊢ (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))})

Distinct variable groups:   𝑥,𝐴   𝑆,𝑜   𝑜,𝑉   𝑥,𝑜,𝑦,𝑊

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

Proof of Theorem lpolsetN

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3478	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lpolset.p	. . 3 ⊢ 𝑃 = (LPol‘𝑊)
3		fveq2 6885	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4		lpolset.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊)
5	3, 4	eqtr4di 2818	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6		fveq2 6885	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7		lpolset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
8	6, 7	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
9	8	pweqd 4581	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
10	5, 9	oveq12d 7437	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) = (𝑆 ↑m 𝒫 𝑉))
11	8	fveq2d 6889	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑜‘(Base‘𝑤)) = (𝑜‘𝑉))
12		fveq2 6885	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
13		lpolset.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑊)
14	12, 13	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
15	14	sneqd 4603	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {(0g‘𝑤)} = { 0 })
16	11, 15	eqeq12d 2781	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ↔ (𝑜‘𝑉) = { 0 }))
17	8	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑥 ⊆ (Base‘𝑤) ↔ 𝑥 ⊆ 𝑉))
18	8	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑦 ⊆ (Base‘𝑤) ↔ 𝑦 ⊆ 𝑉))
19	17, 18	3anbi12d 1465	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) ↔ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)))
20	19	imbi1d 344	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ↔ ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))))
21	20	2albidv 1956	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ↔ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))))
22		fveq2 6885	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSAtoms‘𝑤) = (LSAtoms‘𝑊))
23		lpolset.a	. . . . . . . 8 ⊢ 𝐴 = (LSAtoms‘𝑊)
24	22, 23	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSAtoms‘𝑤) = 𝐴)
25		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSHyp‘𝑤) = (LSHyp‘𝑊))
26		lpolset.h	. . . . . . . . . 10 ⊢ 𝐻 = (LSHyp‘𝑊)
27	25, 26	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSHyp‘𝑤) = 𝐻)
28	27	eleq2d 2851	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ↔ (𝑜‘𝑥) ∈ 𝐻))
29	28	anbi1d 643	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)))
30	24, 29	raleqbidv 3340	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)))
31	16, 21, 30	3anbi123d 1464	. . . . 5 ⊢ (𝑤 = 𝑊 → (((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)) ↔ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))))
32	10, 31	rabeqbidv 3436	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))} = {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))})
33		df-lpolN 42313	. . . 4 ⊢ LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
34		ovex 7452	. . . . 5 ⊢ (𝑆 ↑m 𝒫 𝑉) ∈ V
35	34	rabex 5311	. . . 4 ⊢ {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))} ∈ V
36	32, 33, 35	fvmpt 6993	. . 3 ⊢ (𝑊 ∈ V → (LPol‘𝑊) = {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))})
37	2, 36	eqtrid 2812	. 2 ⊢ (𝑊 ∈ V → 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))})
38	1, 37	syl 18	1 ⊢ (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))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457   ⊆ wss 3906  𝒫 cpw 4564  {csn 4591  ‘cfv 6540  (class class class)co 7419   ↑_m cmap 8830  Basecbs 17291  0_gc0g 17514  LSubSpclss 21102  LSAtomsclsa 39806  LSHypclsh 39807  LPolclpoN 42312

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-lpolN 42313

This theorem is used by:  islpolN  42315