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Definition df-lininds 42754
Description: Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Assertion
Ref Expression
df-lininds 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))))}
Distinct variable group:   𝑓,𝑚,𝑠,𝑥

Detailed syntax breakdown of Definition df-lininds
StepHypRef Expression
1 clininds 42752 . 2 class linIndS
2 vs . . . . . 6 setvar 𝑠
32cv 1630 . . . . 5 class 𝑠
4 vm . . . . . . . 8 setvar 𝑚
54cv 1630 . . . . . . 7 class 𝑚
6 cbs 16064 . . . . . . 7 class Base
75, 6cfv 6030 . . . . . 6 class (Base‘𝑚)
87cpw 4298 . . . . 5 class 𝒫 (Base‘𝑚)
93, 8wcel 2145 . . . 4 wff 𝑠 ∈ 𝒫 (Base‘𝑚)
10 vf . . . . . . . . 9 setvar 𝑓
1110cv 1630 . . . . . . . 8 class 𝑓
12 csca 16152 . . . . . . . . . 10 class Scalar
135, 12cfv 6030 . . . . . . . . 9 class (Scalar‘𝑚)
14 c0g 16308 . . . . . . . . 9 class 0g
1513, 14cfv 6030 . . . . . . . 8 class (0g‘(Scalar‘𝑚))
16 cfsupp 8435 . . . . . . . 8 class finSupp
1711, 15, 16wbr 4787 . . . . . . 7 wff 𝑓 finSupp (0g‘(Scalar‘𝑚))
18 clinc 42716 . . . . . . . . . 10 class linC
195, 18cfv 6030 . . . . . . . . 9 class ( linC ‘𝑚)
2011, 3, 19co 6796 . . . . . . . 8 class (𝑓( linC ‘𝑚)𝑠)
215, 14cfv 6030 . . . . . . . 8 class (0g𝑚)
2220, 21wceq 1631 . . . . . . 7 wff (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)
2317, 22wa 382 . . . . . 6 wff (𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚))
24 vx . . . . . . . . . 10 setvar 𝑥
2524cv 1630 . . . . . . . . 9 class 𝑥
2625, 11cfv 6030 . . . . . . . 8 class (𝑓𝑥)
2726, 15wceq 1631 . . . . . . 7 wff (𝑓𝑥) = (0g‘(Scalar‘𝑚))
2827, 24, 3wral 3061 . . . . . 6 wff 𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))
2923, 28wi 4 . . . . 5 wff ((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)))
3013, 6cfv 6030 . . . . . 6 class (Base‘(Scalar‘𝑚))
31 cmap 8013 . . . . . 6 class 𝑚
3230, 3, 31co 6796 . . . . 5 class ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑠)
3329, 10, 32wral 3061 . . . 4 wff 𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)))
349, 33wa 382 . . 3 wff (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))))
3534, 2, 4copab 4847 . 2 class {⟨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))))}
361, 35wceq 1631 1 wff 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))))}
Colors of variables: wff setvar class
This definition is referenced by:  rellininds  42755  islininds  42758
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