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Theorem linds0 42779
Description: The empty set is always a linearly independet 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 4218 . . . . . 6 𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))
212a1i 12 . . . . 5 (𝑀𝑉 → ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
3 0ex 4925 . . . . . 6 ∅ ∈ V
4 breq1 4790 . . . . . . . . 9 (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5 oveq1 6803 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
65eqeq1d 2773 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g𝑀)))
74, 6anbi12d 616 . . . . . . . 8 (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g𝑀))))
8 fveq1 6332 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓𝑥) = (∅‘𝑥))
98eqeq1d 2773 . . . . . . . . 9 (𝑓 = ∅ → ((𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
109ralbidv 3135 . . . . . . . 8 (𝑓 = ∅ → (∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀)) ↔ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
117, 10imbi12d 333 . . . . . . 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 4357 . . . . . 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 247 . . . 4 (𝑀𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
15 fvex 6344 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
16 map0e 8051 . . . . . . 7 ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = 1𝑜)
1715, 16mp1i 13 . . . . . 6 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = 1𝑜)
18 df1o2 7730 . . . . . 6 1𝑜 = {∅}
1917, 18syl6eq 2821 . . . . 5 (𝑀𝑉 → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = {∅})
2019raleqdv 3293 . . . 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 247 . . 3 (𝑀𝑉 → ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g𝑀)) → ∀𝑥 ∈ ∅ (𝑓𝑥) = (0g‘(Scalar‘𝑀))))
22 0elpw 4966 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2321, 22jctil 509 . 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 2771 . . . 4 (Base‘𝑀) = (Base‘𝑀)
25 eqid 2771 . . . 4 (0g𝑀) = (0g𝑀)
26 eqid 2771 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
27 eqid 2771 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
28 eqid 2771 . . . 4 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
2924, 25, 26, 27, 28islininds 42760 . . 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 670 . 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 247 1 (𝑀𝑉 → ∅ linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  c0 4063  𝒫 cpw 4298  {csn 4317   class class class wbr 4787  cfv 6030  (class class class)co 6796  1𝑜c1o 7710  𝑚 cmap 8013   finSupp cfsupp 8435  Basecbs 16064  Scalarcsca 16152  0gc0g 16308   linC clinc 42718   linIndS clininds 42754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1o 7717  df-map 8015  df-lininds 42756
This theorem is referenced by: (None)
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