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Theorem lsslindf 20088
Description: Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lsslindf.u 𝑈 = (LSubSp‘𝑊)
lsslindf.x 𝑋 = (𝑊s 𝑆)
Assertion
Ref Expression
lsslindf ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))

Proof of Theorem lsslindf
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rellindf 20066 . . . 4 Rel LIndF
21brrelexi 5118 . . 3 (𝐹 LIndF 𝑋𝐹 ∈ V)
32a1i 11 . 2 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 ∈ V))
41brrelexi 5118 . . 3 (𝐹 LIndF 𝑊𝐹 ∈ V)
54a1i 11 . 2 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑊𝐹 ∈ V))
6 simpr 477 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
7 lsslindf.x . . . . . . . . 9 𝑋 = (𝑊s 𝑆)
8 eqid 2621 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
97, 8ressbasss 15853 . . . . . . . 8 (Base‘𝑋) ⊆ (Base‘𝑊)
10 fss 6013 . . . . . . . 8 ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
116, 9, 10sylancl 693 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12 ffn 6002 . . . . . . . . 9 (𝐹:dom 𝐹⟶(Base‘𝑊) → 𝐹 Fn dom 𝐹)
1312adantl 482 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹 Fn dom 𝐹)
14 simp3 1061 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → ran 𝐹𝑆)
15 lsslindf.u . . . . . . . . . . . . 13 𝑈 = (LSubSp‘𝑊)
168, 15lssss 18856 . . . . . . . . . . . 12 (𝑆𝑈𝑆 ⊆ (Base‘𝑊))
17163ad2ant2 1081 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → 𝑆 ⊆ (Base‘𝑊))
187, 8ressbas2 15852 . . . . . . . . . . 11 (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋))
1917, 18syl 17 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → 𝑆 = (Base‘𝑋))
2014, 19sseqtrd 3620 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → ran 𝐹 ⊆ (Base‘𝑋))
2120adantr 481 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → ran 𝐹 ⊆ (Base‘𝑋))
22 df-f 5851 . . . . . . . 8 (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
2313, 21, 22sylanbrc 697 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
2411, 23impbida 876 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
2524adantr 481 . . . . 5 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
26 simpl2 1063 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → 𝑆𝑈)
27 eqid 2621 . . . . . . . . . . . 12 (Scalar‘𝑊) = (Scalar‘𝑊)
287, 27resssca 15952 . . . . . . . . . . 11 (𝑆𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
2928eqcomd 2627 . . . . . . . . . 10 (𝑆𝑈 → (Scalar‘𝑋) = (Scalar‘𝑊))
3026, 29syl 17 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (Scalar‘𝑋) = (Scalar‘𝑊))
3130fveq2d 6152 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
3230fveq2d 6152 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
3332sneqd 4160 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
3431, 33difeq12d 3707 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35 eqid 2621 . . . . . . . . . . . . 13 ( ·𝑠𝑊) = ( ·𝑠𝑊)
367, 35ressvsca 15953 . . . . . . . . . . . 12 (𝑆𝑈 → ( ·𝑠𝑊) = ( ·𝑠𝑋))
3736eqcomd 2627 . . . . . . . . . . 11 (𝑆𝑈 → ( ·𝑠𝑋) = ( ·𝑠𝑊))
3826, 37syl 17 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ( ·𝑠𝑋) = ( ·𝑠𝑊))
3938oveqd 6621 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝑘( ·𝑠𝑋)(𝐹𝑥)) = (𝑘( ·𝑠𝑊)(𝐹𝑥)))
40 simpl1 1062 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → 𝑊 ∈ LMod)
41 imassrn 5436 . . . . . . . . . . . 12 (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ ran 𝐹
42 simpl3 1064 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ran 𝐹𝑆)
4341, 42syl5ss 3594 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44 eqid 2621 . . . . . . . . . . . 12 (LSpan‘𝑊) = (LSpan‘𝑊)
45 eqid 2621 . . . . . . . . . . . 12 (LSpan‘𝑋) = (LSpan‘𝑋)
467, 44, 45, 15lsslsp 18934 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆) → ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4740, 26, 43, 46syl3anc 1323 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4847eqcomd 2627 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4939, 48eleq12d 2692 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5049notbid 308 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (¬ (𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5134, 50raleqbidv 3141 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5251ralbidv 2980 . . . . 5 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5325, 52anbi12d 746 . . . 4 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))) ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
54 ovex 6632 . . . . . . 7 (𝑊s 𝑆) ∈ V
557, 54eqeltri 2694 . . . . . 6 𝑋 ∈ V
5655a1i 11 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → 𝑋 ∈ V)
57 eqid 2621 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
58 eqid 2621 . . . . . 6 ( ·𝑠𝑋) = ( ·𝑠𝑋)
59 eqid 2621 . . . . . 6 (Scalar‘𝑋) = (Scalar‘𝑋)
60 eqid 2621 . . . . . 6 (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
61 eqid 2621 . . . . . 6 (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
6257, 58, 45, 59, 60, 61islindf 20070 . . . . 5 ((𝑋 ∈ V ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ (𝐹:dom 𝐹⟶(Base‘𝑋) ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
6356, 62sylan 488 . . . 4 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ (𝐹:dom 𝐹⟶(Base‘𝑋) ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠𝑋)(𝐹𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
64 eqid 2621 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
65 eqid 2621 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
668, 35, 44, 27, 64, 65islindf 20070 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
67663ad2antl1 1221 . . . 4 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
6853, 63, 673bitr4d 300 . . 3 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))
6968ex 450 . 2 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 ∈ V → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊)))
703, 5, 69pm5.21ndd 369 1 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  {csn 4148   class class class wbr 4613  dom cdm 5074  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  s cress 15782  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  LModclmod 18784  LSubSpclss 18851  LSpanclspn 18890   LIndF clindf 20062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-sca 15878  df-vsca 15879  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-lss 18852  df-lsp 18891  df-lindf 20064
This theorem is referenced by:  lsslinds  20089
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