Theorem lpolsetN 36771

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 3212	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lpolset.p	. . 3 ⊢ 𝑃 = (LPol‘𝑊)
3		fveq2 6191	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4		lpolset.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊)
5	3, 4	syl6eqr 2674	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6		fveq2 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7		lpolset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
8	6, 7	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
9	8	pweqd 4163	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
10	5, 9	oveq12d 6668	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) = (𝑆 ↑𝑚 𝒫 𝑉))
11	8	fveq2d 6195	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑜‘(Base‘𝑤)) = (𝑜‘𝑉))
12		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
13		lpolset.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑊)
14	12, 13	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
15	14	sneqd 4189	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {(0g‘𝑤)} = { 0 })
16	11, 15	eqeq12d 2637	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ↔ (𝑜‘𝑉) = { 0 }))
17	8	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑥 ⊆ (Base‘𝑤) ↔ 𝑥 ⊆ 𝑉))
18	8	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑦 ⊆ (Base‘𝑤) ↔ 𝑦 ⊆ 𝑉))
19	17, 18	3anbi12d 1400	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) ↔ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)))
20	19	imbi1d 331	. . . . . . 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 1851	. . . . . 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 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSAtoms‘𝑤) = (LSAtoms‘𝑊))
23		lpolset.a	. . . . . . . 8 ⊢ 𝐴 = (LSAtoms‘𝑊)
24	22, 23	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSAtoms‘𝑤) = 𝐴)
25		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSHyp‘𝑤) = (LSHyp‘𝑊))
26		lpolset.h	. . . . . . . . . 10 ⊢ 𝐻 = (LSHyp‘𝑊)
27	25, 26	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSHyp‘𝑤) = 𝐻)
28	27	eleq2d 2687	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ↔ (𝑜‘𝑥) ∈ 𝐻))
29	28	anbi1d 741	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)))
30	24, 29	raleqbidv 3152	. . . . . 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 1399	. . . . 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 3195	. . . 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 36770	. . . 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 6678	. . . . 5 ⊢ (𝑆 ↑𝑚 𝒫 𝑉) ∈ V
35	34	rabex 4813	. . . 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 6282	. . 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	syl5eq 2668	. 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 17	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))})

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Basecbs 15857  0_gc0g 16100  LSubSpclss 18932  LSAtomsclsa 34261  LSHypclsh 34262  LPolclpoN 36769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-lpolN 36770

This theorem is referenced by:  islpolN  36772