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Theorem islininds 44914
 Description: The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islininds.b 𝐵 = (Base‘𝑀)
islininds.z 𝑍 = (0g𝑀)
islininds.r 𝑅 = (Scalar‘𝑀)
islininds.e 𝐸 = (Base‘𝑅)
islininds.0 0 = (0g𝑅)
Assertion
Ref Expression
islininds ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,𝑥   𝑆,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝐸(𝑥)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)   0 (𝑥,𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem islininds
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
2 fveq2 6650 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 islininds.b . . . . . . 7 𝐵 = (Base‘𝑀)
42, 3eqtr4di 2851 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
54adantl 485 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘𝑚) = 𝐵)
65pweqd 4516 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
71, 6eleq12d 2884 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵))
8 fveq2 6650 . . . . . . . . 9 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
9 islininds.r . . . . . . . . 9 𝑅 = (Scalar‘𝑀)
108, 9eqtr4di 2851 . . . . . . . 8 (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅)
1110fveq2d 6654 . . . . . . 7 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅))
12 islininds.e . . . . . . 7 𝐸 = (Base‘𝑅)
1311, 12eqtr4di 2851 . . . . . 6 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸)
1413adantl 485 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸)
1514, 1oveq12d 7158 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑m 𝑠) = (𝐸m 𝑆))
168adantl 485 . . . . . . . . . 10 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀))
1716, 9eqtr4di 2851 . . . . . . . . 9 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅)
1817fveq2d 6654 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = (0g𝑅))
19 islininds.0 . . . . . . . 8 0 = (0g𝑅)
2018, 19eqtr4di 2851 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
2120breq2d 5043 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 ))
22 fveq2 6650 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
2322adantl 485 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀))
24 eqidd 2799 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑓 = 𝑓)
2523, 24, 1oveq123d 7161 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆))
26 fveq2 6650 . . . . . . . . 9 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
2726adantl 485 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = (0g𝑀))
28 islininds.z . . . . . . . 8 𝑍 = (0g𝑀)
2927, 28eqtr4di 2851 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = 𝑍)
3025, 29eqeq12d 2814 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓( linC ‘𝑚)𝑠) = (0g𝑚) ↔ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
3121, 30anbi12d 633 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
3210fveq2d 6654 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑅))
3332, 19eqtr4di 2851 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = 0 )
3433adantl 485 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
3534eqeq2d 2809 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ (𝑓𝑥) = 0 ))
361, 35raleqbidv 3354 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥𝑆 (𝑓𝑥) = 0 ))
3731, 36imbi12d 348 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → (((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
3815, 37raleqbidv 3354 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))) ↔ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
397, 38anbi12d 633 . 2 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)))) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
40 df-lininds 44910 . 2 linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))))}
4139, 40brabga 5387 1 ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  𝒫 cpw 4497   class class class wbr 5031  ‘cfv 6327  (class class class)co 7140   ↑m cmap 8396   finSupp cfsupp 8824  Basecbs 16482  Scalarcsca 16567  0gc0g 16712   linC clinc 44872   linIndS clininds 44908 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-iota 6286  df-fv 6335  df-ov 7143  df-lininds 44910 This theorem is referenced by:  linindsi  44915  islinindfis  44917  islindeps  44921  lindslininds  44932  linds0  44933  lindsrng01  44936  snlindsntor  44939
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