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Theorem linds0 44448
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 4452 . . . . . 6 𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))
212a1i 12 . . . . 5 (𝑀𝑉 → ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
3 0ex 5202 . . . . . 6 ∅ ∈ V
4 breq1 5060 . . . . . . . . 9 (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5 oveq1 7152 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
65eqeq1d 2820 . . . . . . . . 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 6662 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓𝑥) = (∅‘𝑥))
98eqeq1d 2820 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
109ralbidv 3194 . . . . . . . 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 4605 . . . . . 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 6676 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
16 map0e 8435 . . . . . . 7 ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
1715, 16mp1i 13 . . . . . 6 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18 df1o2 8105 . . . . . 6 1o = {∅}
1917, 18syl6eq 2869 . . . . 5 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
2019raleqdv 3413 . . . 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 5247 . . 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 2818 . . . 4 (Base‘𝑀) = (Base‘𝑀)
25 eqid 2818 . . . 4 (0g𝑀) = (0g𝑀)
26 eqid 2818 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
27 eqid 2818 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
28 eqid 2818 . . . 4 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
2924, 25, 26, 27, 28islininds 44429 . . 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 1528  wcel 2105  wral 3135  Vcvv 3492  c0 4288  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  cfv 6348  (class class class)co 7145  1oc1o 8084  m cmap 8395   finSupp cfsupp 8821  Basecbs 16471  Scalarcsca 16556  0gc0g 16701   linC clinc 44387   linIndS clininds 44423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1o 8091  df-map 8397  df-lininds 44425
This theorem is referenced by: (None)
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