Theorem islinds2 20659

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 20655	. 2 ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ( I ↾ 𝐹) LIndF 𝑊)))
3	1	fvexi 6513	. . . . . . 7 ⊢ 𝐵 ∈ V
4	3	ssex 5081	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 ∈ V)
5	4	adantl 474	. . . . 5 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → 𝐹 ∈ V)
6		resiexg 7434	. . . . 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 20658	. . . 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 582	. . 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 571	. 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		f1oi 6481	. . . . . . . . 9 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
17		f1of 6444	. . . . . . . . 9 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹⟶𝐹)
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ 𝐹):𝐹⟶𝐹
19		dmresi 5763	. . . . . . . . 9 ⊢ dom ( I ↾ 𝐹) = 𝐹
20	19	feq2i 6336	. . . . . . . 8 ⊢ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹⟶𝐹)
21	18, 20	mpbir 223	. . . . . . 7 ⊢ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹
22		fss 6357	. . . . . . 7 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ∧ 𝐹 ⊆ 𝐵) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
23	21, 22	mpan 677	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
24	23	biantrurd 525	. . . . 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})))))
25	19	raleqi 3353	. . . . . . 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}))))
26		fvresi 6758	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐹 → (( I ↾ 𝐹)‘𝑥) = 𝑥)
27	26	oveq2d 6992	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝑘 · (( I ↾ 𝐹)‘𝑥)) = (𝑘 · 𝑥))
28	19	difeq1i 3985	. . . . . . . . . . . . . . 15 ⊢ (dom ( I ↾ 𝐹) ∖ {𝑥}) = (𝐹 ∖ {𝑥})
29	28	imaeq2i 5768	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥}))
30		difss 3998	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ {𝑥}) ⊆ 𝐹
31		resiima 5784	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∖ {𝑥}) ⊆ 𝐹 → (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥})) = (𝐹 ∖ {𝑥}))
32	30, 31	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥})) = (𝐹 ∖ {𝑥})
33	29, 32	eqtri 2802	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (𝐹 ∖ {𝑥})
34	33	fveq2i 6502	. . . . . . . . . . . 12 ⊢ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) = (𝐾‘(𝐹 ∖ {𝑥}))
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) = (𝐾‘(𝐹 ∖ {𝑥})))
36	27, 35	eleq12d 2860	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐹 → ((𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
37	36	notbid 310	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐹 → (¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
38	37	ralbidv 3147	. . . . . . . 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}))))
39	38	ralbiia 3114	. . . . . . 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})))
40	25, 39	bitri 267	. . . . . 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})))
41	40	anbi2i 613	. . . . 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}))))
42	24, 41	syl6rbbr 282	. . . 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 567	. . 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 297	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 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3088  Vcvv 3415   ∖ cdif 3826   ⊆ wss 3829  {csn 4441   class class class wbr 4929   I cid 5311  dom cdm 5407   ↾ cres 5409   “ cima 5410  ⟶wf 6184  –1-1-onto→wf1o 6187  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  Scalarcsca 16424   ·_𝑠 cvsca 16425  0_gc0g 16569  LSpanclspn 19465   LIndF clindf 20650  LIndSclinds 20651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-lindf 20652  df-linds 20653

This theorem is referenced by:  lindsind  20663  lindfrn  20667  islbs4  20678  0nellinds  30614  lindssn  30616  lindsunlem  30655  lindsun  30656  lindsadd  34332  lindsenlbs  34334  lindslininds  43892