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Theorem lindspropd 32971
Description: Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.)
Hypotheses
Ref Expression
lindfpropd.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
lindfpropd.2 (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))
lindfpropd.3 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
lindfpropd.4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
lindfpropd.5 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
lindfpropd.6 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
lindfpropd.k (𝜑𝐾𝑉)
lindfpropd.l (𝜑𝐿𝑊)
Assertion
Ref Expression
lindspropd (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))
Distinct variable groups:   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem lindspropd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 lindfpropd.1 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
21sseq2d 4007 . . . 4 (𝜑 → (𝑧 ⊆ (Base‘𝐾) ↔ 𝑧 ⊆ (Base‘𝐿)))
3 lindfpropd.2 . . . . 5 (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))
4 lindfpropd.3 . . . . 5 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
5 lindfpropd.4 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
6 lindfpropd.5 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
7 lindfpropd.6 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
8 lindfpropd.k . . . . 5 (𝜑𝐾𝑉)
9 lindfpropd.l . . . . 5 (𝜑𝐿𝑊)
10 vex 3470 . . . . . . 7 𝑧 ∈ V
1110a1i 11 . . . . . 6 (𝜑𝑧 ∈ V)
1211resiexd 7210 . . . . 5 (𝜑 → ( I ↾ 𝑧) ∈ V)
131, 3, 4, 5, 6, 7, 8, 9, 12lindfpropd 32970 . . . 4 (𝜑 → (( I ↾ 𝑧) LIndF 𝐾 ↔ ( I ↾ 𝑧) LIndF 𝐿))
142, 13anbi12d 630 . . 3 (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
15 eqid 2724 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1615islinds 21674 . . . 4 (𝐾𝑉 → (𝑧 ∈ (LIndS‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾)))
178, 16syl 17 . . 3 (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾)))
18 eqid 2724 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
1918islinds 21674 . . . 4 (𝐿𝑊 → (𝑧 ∈ (LIndS‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
209, 19syl 17 . . 3 (𝜑 → (𝑧 ∈ (LIndS‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
2114, 17, 203bitr4d 311 . 2 (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ 𝑧 ∈ (LIndS‘𝐿)))
2221eqrdv 2722 1 (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  wss 3941   class class class wbr 5139   I cid 5564  cres 5669  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  Scalarcsca 17201   ·𝑠 cvsca 17202  0gc0g 17386   LIndF clindf 21669  LIndSclinds 21670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-lss 20771  df-lsp 20811  df-lindf 21671  df-linds 21672
This theorem is referenced by:  fedgmullem2  33197
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