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Theorem linds0 44354
Description: The empty set is always a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
linds0 (𝑀𝑉 → ∅ linIndS 𝑀)

Proof of Theorem linds0
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 4458 . . . . . 6 𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))
212a1i 12 . . . . 5 (𝑀𝑉 → ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
3 0ex 5207 . . . . . 6 ∅ ∈ V
4 breq1 5065 . . . . . . . . 9 (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5 oveq1 7158 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
65eqeq1d 2827 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g𝑀)))
74, 6anbi12d 630 . . . . . . . 8 (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀))))
8 fveq1 6665 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓𝑥) = (∅‘𝑥))
98eqeq1d 2827 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
109ralbidv 3201 . . . . . . . 8 (𝑓 = ∅ → (∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
117, 10imbi12d 346 . . . . . . 7 (𝑓 = ∅ → (((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))) ↔ ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀)))))
1211ralsng 4611 . . . . . 6 (∅ ∈ V → (∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))) ↔ ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀)))))
133, 12mp1i 13 . . . . 5 (𝑀𝑉 → (∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))) ↔ ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀)))))
142, 13mpbird 258 . . . 4 (𝑀𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
15 fvex 6679 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
16 map0e 8439 . . . . . . 7 ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
1715, 16mp1i 13 . . . . . 6 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18 df1o2 8110 . . . . . 6 1o = {∅}
1917, 18syl6eq 2876 . . . . 5 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
2019raleqdv 3420 . . . 4 (𝑀𝑉 → (∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))) ↔ ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀)))))
2114, 20mpbird 258 . . 3 (𝑀𝑉 → ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
22 0elpw 5252 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2321, 22jctil 520 . 2 (𝑀𝑉 → (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀)))))
24 eqid 2825 . . . 4 (Base‘𝑀) = (Base‘𝑀)
25 eqid 2825 . . . 4 (0g𝑀) = (0g𝑀)
26 eqid 2825 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
27 eqid 2825 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
28 eqid 2825 . . . 4 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
2924, 25, 26, 27, 28islininds 44335 . . 3 ((∅ ∈ V ∧ 𝑀𝑉) → (∅ linIndS 𝑀 ↔ (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))))
303, 29mpan 686 . 2 (𝑀𝑉 → (∅ linIndS 𝑀 ↔ (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))))
3123, 30mpbird 258 1 (𝑀𝑉 → ∅ linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499  c0 4294  𝒫 cpw 4541  {csn 4563   class class class wbr 5062  cfv 6351  (class class class)co 7151  1oc1o 8089  m cmap 8399   finSupp cfsupp 8825  Basecbs 16475  Scalarcsca 16560  0gc0g 16705   linC clinc 44293   linIndS clininds 44329
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1o 8096  df-map 8401  df-lininds 44331
This theorem is referenced by: (None)
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