Theorem lssset 19699

Description: The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)

Hypotheses

Ref	Expression
lssset.f	⊢ 𝐹 = (Scalar‘𝑊)
lssset.b	⊢ 𝐵 = (Base‘𝐹)
lssset.v	⊢ 𝑉 = (Base‘𝑊)
lssset.p	⊢ + = (+g‘𝑊)
lssset.t	⊢ · = ( ·𝑠 ‘𝑊)
lssset.s	⊢ 𝑆 = (LSubSp‘𝑊)

Assertion

Ref	Expression
lssset	⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})

Distinct variable groups:   + ,𝑠   𝑥,𝑠,𝐵   𝑉,𝑠   𝑎,𝑏,𝑠,𝑥,𝑊   · ,𝑠

Allowed substitution hints:   𝐵(𝑎,𝑏)   + (𝑥,𝑎,𝑏)   𝑆(𝑥,𝑠,𝑎,𝑏)   · (𝑥,𝑎,𝑏)   𝐹(𝑥,𝑠,𝑎,𝑏)   𝑉(𝑥,𝑎,𝑏)   𝑋(𝑥,𝑠,𝑎,𝑏)

Proof of Theorem lssset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lssset.s	. 2 ⊢ 𝑆 = (LSubSp‘𝑊)
2		elex 3512	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6664	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lssset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	syl6eqr 2874	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4543	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7	6	difeq1d 4097	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝒫 (Base‘𝑤) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
8		fveq2 6664	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		lssset.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
10	8, 9	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
11	10	fveq2d 6668	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
12		lssset.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
13	11, 12	syl6eqr 2874	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐵)
14		fveq2 6664	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
15		lssset.t	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
16	14, 15	syl6eqr 2874	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
17	16	oveqd 7167	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑎) = (𝑥 · 𝑎))
18	17	oveq1d 7165	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎)(+g‘𝑤)𝑏))
19		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = (+g‘𝑊))
20		lssset.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
21	19, 20	syl6eqr 2874	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = + )
22	21	oveqd 7167	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑥 · 𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
23	18, 22	eqtrd 2856	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
24	23	eleq1d 2897	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
25	24	2ralbidv 3199	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
26	13, 25	raleqbidv 3401	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
27	7, 26	rabeqbidv 3485	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠} = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
28		df-lss 19698	. . . 4 ⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠})
29	4	fvexi 6678	. . . . . . 7 ⊢ 𝑉 ∈ V
30	29	pwex 5273	. . . . . 6 ⊢ 𝒫 𝑉 ∈ V
31	30	difexi 5224	. . . . 5 ⊢ (𝒫 𝑉 ∖ {∅}) ∈ V
32	31	rabex 5227	. . . 4 ⊢ {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠} ∈ V
33	27, 28, 32	fvmpt 6762	. . 3 ⊢ (𝑊 ∈ V → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
34	2, 33	syl 17	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
35	1, 34	syl5eq 2868	1 ⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3494   ∖ cdif 3932  ∅c0 4290  𝒫 cpw 4538  {csn 4560  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  Scalarcsca 16562   ·_𝑠 cvsca 16563  LSubSpclss 19697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-lss 19698

This theorem is referenced by:  islss  19700