Theorem lindfind 21771

Description: A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
lindfind.s	⊢ · = ( ·𝑠 ‘𝑊)
lindfind.n	⊢ 𝑁 = (LSpan‘𝑊)
lindfind.l	⊢ 𝐿 = (Scalar‘𝑊)
lindfind.z	⊢ 0 = (0g‘𝐿)
lindfind.k	⊢ 𝐾 = (Base‘𝐿)

Assertion

Ref	Expression
lindfind	⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Proof of Theorem lindfind

Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ dom 𝐹)
2		eldifsn 4742	. . . 4 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	biimpri 228	. . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4	3	adantl 481	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5		simpll 766	. . . 4 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 LIndF 𝑊)
6		lindfind.l	. . . . . . 7 ⊢ 𝐿 = (Scalar‘𝑊)
7		lindfind.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐿)
8	6, 7	elbasfv 17142	. . . . . 6 ⊢ (𝐴 ∈ 𝐾 → 𝑊 ∈ V)
9	8	ad2antrl 728	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝑊 ∈ V)
10		rellindf 21763	. . . . . . 7 ⊢ Rel LIndF
11	10	brrelex1i 5680	. . . . . 6 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
12	11	ad2antrr 726	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 ∈ V)
13		eqid 2736	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
14		lindfind.s	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊)
15		lindfind.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊)
16		lindfind.z	. . . . . 6 ⊢ 0 = (0g‘𝐿)
17	13, 14, 15, 6, 7, 16	islindf 21767	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))))
18	9, 12, 17	syl2anc 584	. . . 4 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))))
19	5, 18	mpbid 232	. . 3 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒})))))
20	19	simprd 495	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))
21		fveq2 6834	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹‘𝑒) = (𝐹‘𝐸))
22	21	oveq2d 7374	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · (𝐹‘𝑒)) = (𝑎 · (𝐹‘𝐸)))
23		sneq 4590	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
24	23	difeq2d 4078	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
25	24	imaeq2d 6019	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
26	25	fveq2d 6838	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
27	22, 26	eleq12d 2830	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
28	27	notbid 318	. . 3 ⊢ (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
29		oveq1 7365	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · (𝐹‘𝐸)) = (𝐴 · (𝐹‘𝐸)))
30	29	eleq1d 2821	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
31	30	notbid 318	. . 3 ⊢ (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
32	28, 31	rspc2va 3588	. 2 ⊢ (((𝐸 ∈ dom 𝐹 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒})))) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
33	1, 4, 20, 32	syl21anc 837	1 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440   ∖ cdif 3898  {csn 4580   class class class wbr 5098  dom cdm 5624   “ cima 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Scalarcsca 17180   ·_𝑠 cvsca 17181  0_gc0g 17359  LSpanclspn 20922   LIndF clindf 21759

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-1cn 11084  ax-addcl 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-nn 12146  df-slot 17109  df-ndx 17121  df-base 17137  df-lindf 21761

This theorem is referenced by:  lindfind2  21773  lindfrn  21776  f1lindf  21777