Theorem islinds2 20152

Description: Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
islindf.b	⊢ 𝐵 = (Base‘𝑊)
islindf.v	⊢ · = ( ·𝑠 ‘𝑊)
islindf.k	⊢ 𝐾 = (LSpan‘𝑊)
islindf.s	⊢ 𝑆 = (Scalar‘𝑊)
islindf.n	⊢ 𝑁 = (Base‘𝑆)
islindf.z	⊢ 0 = (0g‘𝑆)

Assertion

Ref	Expression
islinds2	⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))

Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝑁   𝑘,𝑊,𝑥   0 ,𝑘

Allowed substitution hints:   𝐵(𝑥,𝑘)   𝑆(𝑥,𝑘)   · (𝑥,𝑘)   𝐾(𝑥,𝑘)   𝑁(𝑥)   𝑌(𝑥,𝑘)   0 (𝑥)

Proof of Theorem islinds2

Step	Hyp	Ref	Expression
1		islindf.b	. . 3 ⊢ 𝐵 = (Base‘𝑊)
2	1	islinds 20148	. 2 ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ( I ↾ 𝐹) LIndF 𝑊)))
3		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝑊) ∈ V
4	1, 3	eqeltri 2697	. . . . . . 7 ⊢ 𝐵 ∈ V
5	4	ssex 4802	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 ∈ V)
6	5	adantl 482	. . . . 5 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → 𝐹 ∈ V)
7		resiexg 7102	. . . . 5 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
8	6, 7	syl 17	. . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → ( I ↾ 𝐹) ∈ V)
9		islindf.v	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
10		islindf.k	. . . . 5 ⊢ 𝐾 = (LSpan‘𝑊)
11		islindf.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑊)
12		islindf.n	. . . . 5 ⊢ 𝑁 = (Base‘𝑆)
13		islindf.z	. . . . 5 ⊢ 0 = (0g‘𝑆)
14	1, 9, 10, 11, 12, 13	islindf 20151	. . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ ( I ↾ 𝐹) ∈ V) → (( I ↾ 𝐹) LIndF 𝑊 ↔ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))))))
15	8, 14	syldan 487	. . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → (( I ↾ 𝐹) LIndF 𝑊 ↔ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))))))
16	15	pm5.32da 673	. 2 ⊢ (𝑊 ∈ 𝑌 → ((𝐹 ⊆ 𝐵 ∧ ( I ↾ 𝐹) LIndF 𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})))))))
17		f1oi 6174	. . . . . . . . 9 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
18		f1of 6137	. . . . . . . . 9 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹⟶𝐹)
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ 𝐹):𝐹⟶𝐹
20		dmresi 5457	. . . . . . . . 9 ⊢ dom ( I ↾ 𝐹) = 𝐹
21	20	feq2i 6037	. . . . . . . 8 ⊢ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹⟶𝐹)
22	19, 21	mpbir 221	. . . . . . 7 ⊢ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹
23		fss 6056	. . . . . . 7 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ∧ 𝐹 ⊆ 𝐵) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
24	22, 23	mpan 706	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
25	24	biantrurd 529	. . . . 5 ⊢ (𝐹 ⊆ 𝐵 → (∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})) ↔ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))
26	20	raleqi 3142	. . . . . . 7 ⊢ (∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))))
27		fvresi 6439	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐹 → (( I ↾ 𝐹)‘𝑥) = 𝑥)
28	27	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝑘 · (( I ↾ 𝐹)‘𝑥)) = (𝑘 · 𝑥))
29	20	difeq1i 3724	. . . . . . . . . . . . . . 15 ⊢ (dom ( I ↾ 𝐹) ∖ {𝑥}) = (𝐹 ∖ {𝑥})
30	29	imaeq2i 5464	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥}))
31		difss 3737	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ {𝑥}) ⊆ 𝐹
32		resiima 5480	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∖ {𝑥}) ⊆ 𝐹 → (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥})) = (𝐹 ∖ {𝑥}))
33	31, 32	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥})) = (𝐹 ∖ {𝑥})
34	30, 33	eqtri 2644	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (𝐹 ∖ {𝑥})
35	34	fveq2i 6194	. . . . . . . . . . . 12 ⊢ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) = (𝐾‘(𝐹 ∖ {𝑥}))
36	35	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) = (𝐾‘(𝐹 ∖ {𝑥})))
37	28, 36	eleq12d 2695	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐹 → ((𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
38	37	notbid 308	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐹 → (¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
39	38	ralbidv 2986	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐹 → (∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
40	39	ralbiia 2979	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))
41	26, 40	bitri 264	. . . . . 6 ⊢ (∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))
42	41	anbi2i 730	. . . . 5 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})))) ↔ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
43	25, 42	syl6rbbr 279	. . . 4 ⊢ (𝐹 ⊆ 𝐵 → ((( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})))) ↔ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
44	43	pm5.32i 669	. . 3 ⊢ ((𝐹 ⊆ 𝐵 ∧ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))))) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
45	44	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑌 → ((𝐹 ⊆ 𝐵 ∧ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵 ∧ ∀𝑥 ∈ dom ( I ↾ 𝐹)∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))))) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))
46	2, 16, 45	3bitrd 294	1 ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177   class class class wbr 4653   I cid 5023  dom cdm 5114   ↾ cres 5116   “ cima 5117  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100  LSpanclspn 18971   LIndF clindf 20143  LIndSclinds 20144

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lindf 20145  df-linds 20146

This theorem is referenced by:  lindsind  20156  lindfrn  20160  islbs4  20171  lindsenlbs  33404  lindslininds  42253