Theorem islssm 14037

Description: The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)

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
islssm	⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))

Distinct variable groups:   𝑥,𝐵   𝑎,𝑏,𝑥,𝑊   𝑈,𝑎,𝑏,𝑥,𝑗

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

Proof of Theorem islssm

Dummy variables 𝑠 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lssset.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
2	1	lssmex 14035	. 2 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ V)
3		eleq1w 2265	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑈 ↔ 𝑗 ∈ 𝑈))
4	3	cbvexv 1941	. . . . 5 ⊢ (∃𝑘 𝑘 ∈ 𝑈 ↔ ∃𝑗 𝑗 ∈ 𝑈)
5		ssel 3186	. . . . . . 7 ⊢ (𝑈 ⊆ 𝑉 → (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝑉))
6		lssset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7	6	basmex 12810	. . . . . . 7 ⊢ (𝑘 ∈ 𝑉 → 𝑊 ∈ V)
8	5, 7	syl6 33	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑘 ∈ 𝑈 → 𝑊 ∈ V))
9	8	exlimdv 1841	. . . . 5 ⊢ (𝑈 ⊆ 𝑉 → (∃𝑘 𝑘 ∈ 𝑈 → 𝑊 ∈ V))
10	4, 9	biimtrrid 153	. . . 4 ⊢ (𝑈 ⊆ 𝑉 → (∃𝑗 𝑗 ∈ 𝑈 → 𝑊 ∈ V))
11	10	imp 124	. . 3 ⊢ ((𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈) → 𝑊 ∈ V)
12	11	3adant3 1019	. 2 ⊢ ((𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) → 𝑊 ∈ V)
13		lssset.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
14		lssset.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐹)
15		lssset.p	. . . . 5 ⊢ + = (+g‘𝑊)
16		lssset.t	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
17	13, 14, 6, 15, 16, 1	lsssetm 14036	. . . 4 ⊢ (𝑊 ∈ V → 𝑆 = {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)})
18	17	eleq2d 2274	. . 3 ⊢ (𝑊 ∈ V → (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)}))
19		basfn 12809	. . . . . . . 8 ⊢ Base Fn V
20		funfvex 5587	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
21	20	funfni 5370	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
22	19, 21	mpan 424	. . . . . . 7 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
23	6, 22	eqeltrid 2291	. . . . . 6 ⊢ (𝑊 ∈ V → 𝑉 ∈ V)
24		elpw2g 4199	. . . . . 6 ⊢ (𝑉 ∈ V → (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉))
25	23, 24	syl 14	. . . . 5 ⊢ (𝑊 ∈ V → (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉))
26	25	anbi1d 465	. . . 4 ⊢ (𝑊 ∈ V → ((𝑈 ∈ 𝒫 𝑉 ∧ (∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)) ↔ (𝑈 ⊆ 𝑉 ∧ (∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))))
27		eleq2 2268	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (𝑗 ∈ 𝑠 ↔ 𝑗 ∈ 𝑈))
28	27	exbidv 1847	. . . . . 6 ⊢ (𝑠 = 𝑈 → (∃𝑗 𝑗 ∈ 𝑠 ↔ ∃𝑗 𝑗 ∈ 𝑈))
29		eleq2 2268	. . . . . . . . 9 ⊢ (𝑠 = 𝑈 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
30	29	raleqbi1dv 2713	. . . . . . . 8 ⊢ (𝑠 = 𝑈 → (∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
31	30	raleqbi1dv 2713	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
32	31	ralbidv 2505	. . . . . 6 ⊢ (𝑠 = 𝑈 → (∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
33	28, 32	anbi12d 473	. . . . 5 ⊢ (𝑠 = 𝑈 → ((∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠) ↔ (∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)))
34	33	elrab 2928	. . . 4 ⊢ (𝑈 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)} ↔ (𝑈 ∈ 𝒫 𝑉 ∧ (∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)))
35		3anass 984	. . . 4 ⊢ ((𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) ↔ (𝑈 ⊆ 𝑉 ∧ (∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)))
36	26, 34, 35	3bitr4g 223	. . 3 ⊢ (𝑊 ∈ V → (𝑈 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)} ↔ (𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)))
37	18, 36	bitrd 188	. 2 ⊢ (𝑊 ∈ V → (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)))
38	2, 12, 37	pm5.21nii 705	1 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∀wral 2483  {crab 2487  Vcvv 2771   ⊆ wss 3165  𝒫 cpw 3615   Fn wfn 5263  ‘cfv 5268  (class class class)co 5934  Basecbs 12751  +_gcplusg 12828  Scalarcsca 12831   ·_𝑠 cvsca 12832  LSubSpclss 14032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fn 5271  df-fv 5276  df-ov 5937  df-inn 9019  df-ndx 12754  df-slot 12755  df-base 12757  df-lssm 14033

This theorem is referenced by:  islidlm  14159