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Theorem linds0 46536
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 4470 . . . . . 6 𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))
212a1i 12 . . . . 5 (𝑀𝑉 → ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
3 0ex 5264 . . . . . 6 ∅ ∈ V
4 breq1 5108 . . . . . . . . 9 (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5 oveq1 7364 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
65eqeq1d 2738 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g𝑀)))
74, 6anbi12d 631 . . . . . . . 8 (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀))))
8 fveq1 6841 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓𝑥) = (∅‘𝑥))
98eqeq1d 2738 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
109ralbidv 3174 . . . . . . . 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 4634 . . . . . 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 256 . . . 4 (𝑀𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
15 fvex 6855 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
16 map0e 8820 . . . . . . 7 ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
1715, 16mp1i 13 . . . . . 6 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18 df1o2 8419 . . . . . 6 1o = {∅}
1917, 18eqtrdi 2792 . . . . 5 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
2019raleqdv 3313 . . . 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 256 . . 3 (𝑀𝑉 → ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
22 0elpw 5311 . . 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 2736 . . . 4 (Base‘𝑀) = (Base‘𝑀)
25 eqid 2736 . . . 4 (0g𝑀) = (0g𝑀)
26 eqid 2736 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
27 eqid 2736 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
28 eqid 2736 . . . 4 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
2924, 25, 26, 27, 28islininds 46517 . . 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 688 . 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 256 1 (𝑀𝑉 → ∅ linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  c0 4282  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cfv 6496  (class class class)co 7357  1oc1o 8405  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  Scalarcsca 17136  0gc0g 17321   linC clinc 46475   linIndS clininds 46511
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1o 8412  df-map 8767  df-lininds 46513
This theorem is referenced by: (None)
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