Theorem islinds2 20957

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 20953	. 2 ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ( I ↾ 𝐹) LIndF 𝑊)))
3	1	fvexi 6684	. . . . . . 7 ⊢ 𝐵 ∈ V
4	3	ssex 5225	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 ∈ V)
5	4	adantl 484	. . . . 5 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ⊆ 𝐵) → 𝐹 ∈ V)
6		resiexg 7619	. . . . 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 20956	. . . 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 593	. . 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 581	. 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 6652	. . . . . . . . 9 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
17		f1of 6615	. . . . . . . . 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 5921	. . . . . . . . 9 ⊢ dom ( I ↾ 𝐹) = 𝐹
20	19	feq2i 6506	. . . . . . . 8 ⊢ (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹⟶𝐹)
21	18, 20	mpbir 233	. . . . . . 7 ⊢ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹
22		fss 6527	. . . . . . 7 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ∧ 𝐹 ⊆ 𝐵) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
23	21, 22	mpan 688	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐵)
24	23	biantrurd 535	. . . . 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 3413	. . . . . . 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 6935	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐹 → (( I ↾ 𝐹)‘𝑥) = 𝑥)
27	26	oveq2d 7172	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → (𝑘 · (( I ↾ 𝐹)‘𝑥)) = (𝑘 · 𝑥))
28	19	difeq1i 4095	. . . . . . . . . . . . . . 15 ⊢ (dom ( I ↾ 𝐹) ∖ {𝑥}) = (𝐹 ∖ {𝑥})
29	28	imaeq2i 5927	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (( I ↾ 𝐹) “ (𝐹 ∖ {𝑥}))
30		difss 4108	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ {𝑥}) ⊆ 𝐹
31		resiima 5944	. . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥})) = (𝐹 ∖ {𝑥})
34	33	fveq2i 6673	. . . . . . . . . . . 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 2907	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐹 → ((𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
37	36	notbid 320	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐹 → (¬ (𝑘 · (( I ↾ 𝐹)‘𝑥)) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝑥}))) ↔ ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))
38	37	ralbidv 3197	. . . . . . . 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 3164	. . . . . . 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 277	. . . . . 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 624	. . . . 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 292	. . . 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 577	. . 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 307	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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936  {csn 4567   class class class wbr 5066   I cid 5459  dom cdm 5555   ↾ cres 5557   “ cima 5558  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Scalarcsca 16568   ·_𝑠 cvsca 16569  0_gc0g 16713  LSpanclspn 19743   LIndF clindf 20948  LIndSclinds 20949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-lindf 20950  df-linds 20951

This theorem is referenced by:  lindsind  20961  lindfrn  20965  islbs4  20976  0nellinds  30935  lindssn  30939  lindsunlem  31020  lindsun  31021  lindsadd  34900  lindsenlbs  34902  lindslininds  44539