Theorem islssm 13913

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 13911	. 2 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ V)
3		eleq1w 2257	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑈 ↔ 𝑗 ∈ 𝑈))
4	3	cbvexv 1933	. . . . 5 ⊢ (∃𝑘 𝑘 ∈ 𝑈 ↔ ∃𝑗 𝑗 ∈ 𝑈)
5		ssel 3177	. . . . . . 7 ⊢ (𝑈 ⊆ 𝑉 → (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝑉))
6		lssset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7	6	basmex 12737	. . . . . . 7 ⊢ (𝑘 ∈ 𝑉 → 𝑊 ∈ V)
8	5, 7	syl6 33	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑘 ∈ 𝑈 → 𝑊 ∈ V))
9	8	exlimdv 1833	. . . . 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 13912	. . . 4 ⊢ (𝑊 ∈ V → 𝑆 = {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)})
18	17	eleq2d 2266	. . 3 ⊢ (𝑊 ∈ V → (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)}))
19		basfn 12736	. . . . . . . 8 ⊢ Base Fn V
20		funfvex 5575	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
21	20	funfni 5358	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
22	19, 21	mpan 424	. . . . . . 7 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
23	6, 22	eqeltrid 2283	. . . . . 6 ⊢ (𝑊 ∈ V → 𝑉 ∈ V)
24		elpw2g 4189	. . . . . 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 2260	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (𝑗 ∈ 𝑠 ↔ 𝑗 ∈ 𝑈))
28	27	exbidv 1839	. . . . . 6 ⊢ (𝑠 = 𝑈 → (∃𝑗 𝑗 ∈ 𝑠 ↔ ∃𝑗 𝑗 ∈ 𝑈))
29		eleq2 2260	. . . . . . . . 9 ⊢ (𝑠 = 𝑈 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
30	29	raleqbi1dv 2705	. . . . . . . 8 ⊢ (𝑠 = 𝑈 → (∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
31	30	raleqbi1dv 2705	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠 ↔ ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
32	31	ralbidv 2497	. . . . . 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 2920	. . . 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 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  {crab 2479  Vcvv 2763   ⊆ wss 3157  𝒫 cpw 3605   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  Scalarcsca 12758   ·_𝑠 cvsca 12759  LSubSpclss 13908

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-lssm 13909

This theorem is referenced by:  islidlm  14035