Theorem islinds2 21747

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 21743	. 2 ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ( I ↾ 𝐹) LIndF 𝑊)))
3	1	fvexi 6911	. . . . . . 7 ⊢ 𝐵 ∈ V
4	3	ssex 5321	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 ∈ V)
5	4	adantl 481	. . . . 5 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → 𝐹 ∈ V)
6		resiexg 7920	. . . . 5 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → ( I ↾ 𝐹) ∈ V)
8		islindf.v	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
9		islindf.k	. . . . 5 ⊢ 𝐾 = (LSpan‘𝑊)
10		islindf.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑊)
11		islindf.n	. . . . 5 ⊢ 𝑁 = (Base‘𝑆)
12		islindf.z	. . . . 5 ⊢ 0 = (0g‘𝑆)
13	1, 8, 9, 10, 11, 12	islindf 21746	. . . 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}))))))
14	7, 13	syldan 590	. . 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}))))))
15	14	pm5.32da 578	. 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})))))))
16		dmresi 6055	. . . . . . . 8 ⊢ dom ( I ↾ 𝐹) = 𝐹
17	16	raleqi 3320	. . . . . . 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}))))
18		fvresi 7182	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐹 → (( I ↾ 𝐹)‘𝑥) = 𝑥)
19	18	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝑘 · (( I ↾ 𝐹)‘𝑥)) = (𝑘 · 𝑥))
20	16	difeq1i 4116	. . . . . . . . . . . . . . 15 ⊢ (dom ( I ↾ 𝐹) ∖ {𝑥}) = (𝐹 ∖ {𝑥})
21	20	imaeq2i 6061	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥}))
22		difss 4130	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ {𝑥}) ⊆ 𝐹
23		resiima 6079	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∖ {𝑥}) ⊆ 𝐹 → (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥})) = (𝐹 ∖ {𝑥}))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥})) = (𝐹 ∖ {𝑥})
25	21, 24	eqtri 2756	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (𝐹 ∖ {𝑥})
26	25	fveq2i 6900	. . . . . . . . . . . 12 ⊢ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) = (𝐾‘(𝐹 ∖ {𝑥}))
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) = (𝐾‘(𝐹 ∖ {𝑥})))
28	19, 27	eleq12d 2823	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐹 → ((𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
29	28	notbid 318	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐹 → (¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
30	29	ralbidv 3174	. . . . . . . 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}))))
31	30	ralbiia 3088	. . . . . . 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})))
32	17, 31	bitri 275	. . . . . 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})))
33	32	anbi2i 622	. . . . 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}))))
34		f1oi 6877	. . . . . . . . 9 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
35		f1of 6839	. . . . . . . . 9 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹⟶𝐹)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ 𝐹):𝐹⟶𝐹
37	16	feq2i 6714	. . . . . . . 8 ⊢ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹⟶𝐹)
38	36, 37	mpbir 230	. . . . . . 7 ⊢ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹
39		fss 6739	. . . . . . 7 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ∧ 𝐹 ⊆ 𝐵) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
40	38, 39	mpan 689	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
41	40	biantrurd 532	. . . . 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})))))
42	33, 41	bitr4id 290	. . . 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}))))
43	42	pm5.32i 574	. . 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}))))
44	43	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})))))
45	2, 15, 44	3bitrd 305	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 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471   ∖ cdif 3944   ⊆ wss 3947  {csn 4629   class class class wbr 5148   I cid 5575  dom cdm 5678   ↾ cres 5680   “ cima 5681  ⟶wf 6544  –1-1-onto→wf1o 6547  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  Scalarcsca 17236   ·_𝑠 cvsca 17237  0_gc0g 17421  LSpanclspn 20855   LIndF clindf 21738  LIndSclinds 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-lindf 21740  df-linds 21741

This theorem is referenced by:  lindsind  21751  lindfrn  21755  islbs4  21766  0nellinds  33095  lindssn  33106  lindsunlem  33322  lindsun  33323  lindsadd  37086  lindsenlbs  37088  lindslininds  47532