Theorem lindfind 21854

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 769	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ dom 𝐹)
2		eldifsn 4791	. . . 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 767	. . . 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 17251	. . . . . 6 ⊢ (𝐴 ∈ 𝐾 → 𝑊 ∈ V)
9	8	ad2antrl 728	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝑊 ∈ V)
10		rellindf 21846	. . . . . . 7 ⊢ Rel LIndF
11	10	brrelex1i 5745	. . . . . 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 2735	. . . . . 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 21850	. . . . 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 6907	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹‘𝑒) = (𝐹‘𝐸))
22	21	oveq2d 7447	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · (𝐹‘𝑒)) = (𝑎 · (𝐹‘𝐸)))
23		sneq 4641	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
24	23	difeq2d 4136	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
25	24	imaeq2d 6080	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
26	25	fveq2d 6911	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
27	22, 26	eleq12d 2833	. . . 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 7438	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · (𝐹‘𝐸)) = (𝐴 · (𝐹‘𝐸)))
30	29	eleq1d 2824	. . . 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 3634	. 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 838	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 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  Vcvv 3478   ∖ cdif 3960  {csn 4631   class class class wbr 5148  dom cdm 5689   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·_𝑠 cvsca 17302  0_gc0g 17486  LSpanclspn 20987   LIndF clindf 21842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-lindf 21844

This theorem is referenced by:  lindfind2  21856  lindfrn  21859  f1lindf  21860