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Theorem linds0 48382
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 4513 . . . . . 6 𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))
212a1i 12 . . . . 5 (𝑀𝑉 → ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
3 0ex 5307 . . . . . 6 ∅ ∈ V
4 breq1 5146 . . . . . . . . 9 (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5 oveq1 7438 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
65eqeq1d 2739 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g𝑀)))
74, 6anbi12d 632 . . . . . . . 8 (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀))))
8 fveq1 6905 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓𝑥) = (∅‘𝑥))
98eqeq1d 2739 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
109ralbidv 3178 . . . . . . . 8 (𝑓 = ∅ → (∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
117, 10imbi12d 344 . . . . . . 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 4675 . . . . . 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 257 . . . 4 (𝑀𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
15 fvex 6919 . . . . . 6 (Base‘(Scalar‘𝑀)) ∈ V
16 map0e 8922 . . . . . 6 ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
1715, 16mp1i 13 . . . . 5 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18 df1o2 8513 . . . . 5 1o = {∅}
1917, 18eqtrdi 2793 . . . 4 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
2014, 19raleqtrrdv 3330 . . 3 (𝑀𝑉 → ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
21 0elpw 5356 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2220, 21jctil 519 . 2 (𝑀𝑉 → (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀)))))
23 eqid 2737 . . . 4 (Base‘𝑀) = (Base‘𝑀)
24 eqid 2737 . . . 4 (0g𝑀) = (0g𝑀)
25 eqid 2737 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
26 eqid 2737 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27 eqid 2737 . . . 4 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
2823, 24, 25, 26, 27islininds 48363 . . 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))))))
293, 28mpan 690 . 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))))))
3022, 29mpbird 257 1 (𝑀𝑉 → ∅ linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431  1oc1o 8499  m cmap 8866   finSupp cfsupp 9401  Basecbs 17247  Scalarcsca 17300  0gc0g 17484   linC clinc 48321   linIndS clininds 48357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-map 8868  df-lininds 48359
This theorem is referenced by: (None)
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