Theorem lindfind 21795

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 775	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ dom 𝐹)
2		eldifsn 4722	. . 3 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	bilanri 508	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4		simpll 773	. . . 4 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 LIndF 𝑊)
5		lindfind.l	. . . . . . 7 ⊢ 𝐿 = (Scalar‘𝑊)
6		lindfind.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐿)
7	5, 6	elbasfv 17180	. . . . . 6 ⊢ (𝐴 ∈ 𝐾 → 𝑊 ∈ V)
8	7	ad2antrl 735	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝑊 ∈ V)
9		rellindf 21787	. . . . . . 7 ⊢ Rel LIndF
10	9	brrelex1i 5677	. . . . . 6 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
11	10	ad2antrr 733	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 ∈ V)
12		eqid 2741	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
13		lindfind.s	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊)
14		lindfind.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊)
15		lindfind.z	. . . . . 6 ⊢ 0 = (0g‘𝐿)
16	12, 13, 14, 5, 6, 15	islindf 21791	. . . . 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}))))))
17	8, 11, 16	syl2anc 591	. . . 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}))))))
18	4, 17	mpbid 234	. . 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})))))
19	18	simprd 497	. 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}))))
20		fveq2 6831	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹‘𝑒) = (𝐹‘𝐸))
21	20	oveq2d 7376	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · (𝐹‘𝑒)) = (𝑎 · (𝐹‘𝐸)))
22		sneq 4568	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
23	22	difeq2d 4060	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
24	23	imaeq2d 6019	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
25	24	fveq2d 6835	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
26	21, 25	eleq12d 2835	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
27	26	notbid 320	. . 3 ⊢ (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
28		oveq1 7367	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · (𝐹‘𝐸)) = (𝐴 · (𝐹‘𝐸)))
29	28	eleq1d 2826	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
30	29	notbid 320	. . 3 ⊢ (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
31	27, 30	rspc2va 3574	. 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}))))
32	1, 3, 19, 31	syl21anc 844	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 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  Vcvv 3433   ∖ cdif 3882  {csn 4558   class class class wbr 5075  dom cdm 5621   “ cima 5624  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  Scalarcsca 17218   ·_𝑠 cvsca 17219  0_gc0g 17397  LSpanclspn 20965   LIndF clindf 21783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-1cn 11091  ax-addcl 11093

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-nn 12170  df-slot 17147  df-ndx 17159  df-base 17175  df-lindf 21785

This theorem is referenced by:  lindfind2  21797  lindfrn  21800  f1lindf  21801