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Theorem lindfpropd 31000
Description: Property deduction for linearly independent families. (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)
lindfpropd.x (𝜑𝑋𝐴)
Assertion
Ref Expression
lindfpropd (𝜑 → (𝑋 LIndF 𝐾𝑋 LIndF 𝐿))
Distinct variable groups:   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑋,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem lindfpropd
Dummy variables 𝑖 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lindfpropd.2 . . . . . . . 8 (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))
2 lindfpropd.3 . . . . . . . . 9 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
32sneqd 4540 . . . . . . . 8 (𝜑 → {(0g‘(Scalar‘𝐾))} = {(0g‘(Scalar‘𝐿))})
41, 3difeq12d 4054 . . . . . . 7 (𝜑 → ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) = ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}))
54ad2antrr 725 . . . . . 6 (((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) = ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}))
6 simplll 774 . . . . . . . . 9 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → 𝜑)
7 simpr 488 . . . . . . . . . 10 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}))
87eldifad 3896 . . . . . . . . 9 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → 𝑘 ∈ (Base‘(Scalar‘𝐾)))
9 simpr 488 . . . . . . . . . . 11 ((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) → 𝑋:dom 𝑋⟶(Base‘𝐾))
109ffvelrnda 6832 . . . . . . . . . 10 (((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → (𝑋𝑖) ∈ (Base‘𝐾))
1110adantr 484 . . . . . . . . 9 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → (𝑋𝑖) ∈ (Base‘𝐾))
12 lindfpropd.6 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
1312oveqrspc2v 7166 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝐾)) ∧ (𝑋𝑖) ∈ (Base‘𝐾))) → (𝑘( ·𝑠𝐾)(𝑋𝑖)) = (𝑘( ·𝑠𝐿)(𝑋𝑖)))
146, 8, 11, 13syl12anc 835 . . . . . . . 8 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → (𝑘( ·𝑠𝐾)(𝑋𝑖)) = (𝑘( ·𝑠𝐿)(𝑋𝑖)))
15 eqidd 2802 . . . . . . . . . . 11 (𝜑 → (Base‘𝐾) = (Base‘𝐾))
16 lindfpropd.1 . . . . . . . . . . 11 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
17 ssidd 3941 . . . . . . . . . . 11 (𝜑 → (Base‘𝐾) ⊆ (Base‘𝐾))
18 lindfpropd.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
19 lindfpropd.5 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))
20 eqidd 2802 . . . . . . . . . . 11 (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾)))
21 lindfpropd.k . . . . . . . . . . 11 (𝜑𝐾𝑉)
22 lindfpropd.l . . . . . . . . . . 11 (𝜑𝐿𝑊)
2315, 16, 17, 18, 19, 12, 20, 1, 21, 22lsppropd 19787 . . . . . . . . . 10 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
2423fveq1d 6651 . . . . . . . . 9 (𝜑 → ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) = ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))
2524ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) = ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))
2614, 25eleq12d 2887 . . . . . . 7 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → ((𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
2726notbid 321 . . . . . 6 ((((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → (¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
285, 27raleqbidva 3373 . . . . 5 (((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → (∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
2928ralbidva 3164 . . . 4 ((𝜑𝑋:dom 𝑋⟶(Base‘𝐾)) → (∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
3029pm5.32da 582 . . 3 (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))) ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
3116feq3d 6478 . . . 4 (𝜑 → (𝑋:dom 𝑋⟶(Base‘𝐾) ↔ 𝑋:dom 𝑋⟶(Base‘𝐿)))
3231anbi1d 632 . . 3 (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))) ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
3330, 32bitrd 282 . 2 (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))) ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
34 lindfpropd.x . . 3 (𝜑𝑋𝐴)
35 eqid 2801 . . . 4 (Base‘𝐾) = (Base‘𝐾)
36 eqid 2801 . . . 4 ( ·𝑠𝐾) = ( ·𝑠𝐾)
37 eqid 2801 . . . 4 (LSpan‘𝐾) = (LSpan‘𝐾)
38 eqid 2801 . . . 4 (Scalar‘𝐾) = (Scalar‘𝐾)
39 eqid 2801 . . . 4 (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
40 eqid 2801 . . . 4 (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
4135, 36, 37, 38, 39, 40islindf 20505 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑋 LIndF 𝐾 ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
4221, 34, 41syl2anc 587 . 2 (𝜑 → (𝑋 LIndF 𝐾 ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠𝐾)(𝑋𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
43 eqid 2801 . . . 4 (Base‘𝐿) = (Base‘𝐿)
44 eqid 2801 . . . 4 ( ·𝑠𝐿) = ( ·𝑠𝐿)
45 eqid 2801 . . . 4 (LSpan‘𝐿) = (LSpan‘𝐿)
46 eqid 2801 . . . 4 (Scalar‘𝐿) = (Scalar‘𝐿)
47 eqid 2801 . . . 4 (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
48 eqid 2801 . . . 4 (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
4943, 44, 45, 46, 47, 48islindf 20505 . . 3 ((𝐿𝑊𝑋𝐴) → (𝑋 LIndF 𝐿 ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
5022, 34, 49syl2anc 587 . 2 (𝜑 → (𝑋 LIndF 𝐿 ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠𝐿)(𝑋𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
5133, 42, 503bitr4d 314 1 (𝜑 → (𝑋 LIndF 𝐾𝑋 LIndF 𝐿))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  cdif 3881  {csn 4528   class class class wbr 5033  dom cdm 5523  cima 5526  wf 6324  cfv 6328  (class class class)co 7139  Basecbs 16479  +gcplusg 16561  Scalarcsca 16564   ·𝑠 cvsca 16565  0gc0g 16709  LSpanclspn 19740   LIndF clindf 20497
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-lss 19701  df-lsp 19741  df-lindf 20499
This theorem is referenced by:  lindspropd  31001
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