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Theorem lsslindf 20974
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 20952 . . . 4 Rel LIndF
21brrelex1i 5595 . . 3 (𝐹 LIndF 𝑋𝐹 ∈ V)
32a1i 11 . 2 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 ∈ V))
41brrelex1i 5595 . . 3 (𝐹 LIndF 𝑊𝐹 ∈ V)
54a1i 11 . 2 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑊𝐹 ∈ V))
6 simpr 488 . . . . . . . 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 2824 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
97, 8ressbasss 16556 . . . . . . . 8 (Base‘𝑋) ⊆ (Base‘𝑊)
10 fss 6517 . . . . . . . 8 ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
116, 9, 10sylancl 589 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12 ffn 6503 . . . . . . . . 9 (𝐹:dom 𝐹⟶(Base‘𝑊) → 𝐹 Fn dom 𝐹)
1312adantl 485 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹 Fn dom 𝐹)
14 simp3 1135 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → ran 𝐹𝑆)
15 lsslindf.u . . . . . . . . . . . . 13 𝑈 = (LSubSp‘𝑊)
168, 15lssss 19708 . . . . . . . . . . . 12 (𝑆𝑈𝑆 ⊆ (Base‘𝑊))
17163ad2ant2 1131 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → 𝑆 ⊆ (Base‘𝑊))
187, 8ressbas2 16555 . . . . . . . . . . 11 (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋))
1917, 18syl 17 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → 𝑆 = (Base‘𝑋))
2014, 19sseqtrd 3993 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → ran 𝐹 ⊆ (Base‘𝑋))
2120adantr 484 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → ran 𝐹 ⊆ (Base‘𝑋))
22 df-f 6347 . . . . . . . 8 (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
2313, 21, 22sylanbrc 586 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
2411, 23impbida 800 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
2524adantr 484 . . . . 5 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
26 simpl2 1189 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → 𝑆𝑈)
27 eqid 2824 . . . . . . . . . . . 12 (Scalar‘𝑊) = (Scalar‘𝑊)
287, 27resssca 16650 . . . . . . . . . . 11 (𝑆𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
2928eqcomd 2830 . . . . . . . . . 10 (𝑆𝑈 → (Scalar‘𝑋) = (Scalar‘𝑊))
3026, 29syl 17 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (Scalar‘𝑋) = (Scalar‘𝑊))
3130fveq2d 6665 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
3230fveq2d 6665 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
3332sneqd 4562 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
3431, 33difeq12d 4086 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35 eqid 2824 . . . . . . . . . . . . 13 ( ·𝑠𝑊) = ( ·𝑠𝑊)
367, 35ressvsca 16651 . . . . . . . . . . . 12 (𝑆𝑈 → ( ·𝑠𝑊) = ( ·𝑠𝑋))
3736eqcomd 2830 . . . . . . . . . . 11 (𝑆𝑈 → ( ·𝑠𝑋) = ( ·𝑠𝑊))
3826, 37syl 17 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ( ·𝑠𝑋) = ( ·𝑠𝑊))
3938oveqd 7166 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝑘( ·𝑠𝑋)(𝐹𝑥)) = (𝑘( ·𝑠𝑊)(𝐹𝑥)))
40 simpl1 1188 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → 𝑊 ∈ LMod)
41 imassrn 5927 . . . . . . . . . . . 12 (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ ran 𝐹
42 simpl3 1190 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ran 𝐹𝑆)
4341, 42sstrid 3964 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44 eqid 2824 . . . . . . . . . . . 12 (LSpan‘𝑊) = (LSpan‘𝑊)
45 eqid 2824 . . . . . . . . . . . 12 (LSpan‘𝑋) = (LSpan‘𝑋)
467, 44, 45, 15lsslsp 19787 . . . . . . . . . . 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 1368 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4847eqcomd 2830 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4939, 48eleq12d 2910 . . . . . . . 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 321 . . . . . . 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 3392 . . . . . 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 3192 . . . . 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 633 . . . 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}))))))
547ovexi 7183 . . . . . 6 𝑋 ∈ V
5554a1i 11 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → 𝑋 ∈ V)
56 eqid 2824 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
57 eqid 2824 . . . . . 6 ( ·𝑠𝑋) = ( ·𝑠𝑋)
58 eqid 2824 . . . . . 6 (Scalar‘𝑋) = (Scalar‘𝑋)
59 eqid 2824 . . . . . 6 (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
60 eqid 2824 . . . . . 6 (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
6156, 57, 45, 58, 59, 60islindf 20956 . . . . 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}))))))
6255, 61sylan 583 . . . 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}))))))
63 eqid 2824 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
64 eqid 2824 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
658, 35, 44, 27, 63, 64islindf 20956 . . . . 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}))))))
66653ad2antl1 1182 . . . 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}))))))
6753, 62, 663bitr4d 314 . . 3 (((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))
6867ex 416 . 2 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 ∈ V → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊)))
693, 5, 68pm5.21ndd 384 1 ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cdif 3916  wss 3919  {csn 4550   class class class wbr 5052  dom cdm 5542  ran crn 5543  cima 5545   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  s cress 16484  Scalarcsca 16568   ·𝑠 cvsca 16569  0gc0g 16713  LModclmod 19634  LSubSpclss 19703  LSpanclspn 19743   LIndF clindf 20948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-sca 16581  df-vsca 16582  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-sbg 18108  df-subg 18276  df-mgp 19240  df-ur 19252  df-ring 19299  df-lmod 19636  df-lss 19704  df-lsp 19744  df-lindf 20950
This theorem is referenced by:  lsslinds  20975
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