Theorem el0ldep 49404

Description: A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
el0ldep	⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Proof of Theorem el0ldep

Dummy variables 𝑓 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2762	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2762	. . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4		eqid 2762	. . . . 5 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5		eqeq1 2766	. . . . . . 7 ⊢ (𝑠 = 𝑦 → (𝑠 = (0g‘𝑀) ↔ 𝑦 = (0g‘𝑀)))
6	5	ifbid 4509	. . . . . 6 ⊢ (𝑠 = 𝑦 → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
7	6	cbvmptv 5213	. . . . 5 ⊢ (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) = (𝑦 ∈ 𝑆 ↦ if(𝑦 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
8	1, 2, 3, 4, 7	mptcfsupp 49315	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
9	8	3adant1r 1196	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
10		simp1l 1216	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11		simp2 1155	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12		eqid 2762	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
13		eqid 2762	. . . . 5 ⊢ (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
14	1, 2, 3, 4, 12, 13	linc0scn0 49361	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g‘𝑀))
15	10, 11, 14	syl2anc 596	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g‘𝑀))
16		simp3 1156	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (0g‘𝑀) ∈ 𝑆)
17		fveq2 6882	. . . . . 6 ⊢ (𝑥 = (0g‘𝑀) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) = ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g‘𝑀)))
18	17	neeq1d 3016	. . . . 5 ⊢ (𝑥 = (0g‘𝑀) → (((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g‘𝑀)) ≠ (0g‘(Scalar‘𝑀))))
19	18	adantl 487	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) ∧ 𝑥 = (0g‘𝑀)) → (((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g‘𝑀)) ≠ (0g‘(Scalar‘𝑀))))
20		iftrue 4491	. . . . . 6 ⊢ (𝑠 = (0g‘𝑀) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21		fvexd 6897	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
22	13, 20, 16, 21	fvmptd3 7014	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g‘𝑀)) = (1r‘(Scalar‘𝑀)))
23	2	lmodring 21058	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
24	23	anim1i 627	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
25	24	3ad2ant1 1151	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
26		eqid 2762	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27	26, 4, 3	ring1ne0 20447	. . . . . 6 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
28	25, 27	syl 18	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
29	22, 28	eqnetrd 3024	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
30	16, 19, 29	rspcedvd 3581	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ∃𝑥 ∈ 𝑆 ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)))
31	2, 26, 4	lmod1cl 21079	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
32	2, 26, 3	lmod0cl 21078	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
33	31, 32	ifcld 4532	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
34	33	adantr 486	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
35	34	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
36	35	adantr 486	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) ∧ 𝑠 ∈ 𝑆) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
37	36	fmpttd 7112	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑆⟶(Base‘(Scalar‘𝑀)))
38		fvexd 6897	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
39	38, 11	elmapd 8843	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆) ↔ (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑆⟶(Base‘(Scalar‘𝑀))))
40	37, 39	mpbird 260	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆))
41		breq1 5110	. . . . . 6 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀))))
42		oveq1 7424	. . . . . . 7 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆))
43	42	eqeq1d 2764	. . . . . 6 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g‘𝑀)))
44		fveq1 6881	. . . . . . . 8 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓‘𝑥) = ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥))
45	44	neeq1d 3016	. . . . . . 7 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀))))
46	45	rexbidv 3188	. . . . . 6 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ∃𝑥 ∈ 𝑆 ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀))))
47	41, 43, 46	3anbi123d 1464	. . . . 5 ⊢ (𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀))) ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
48	47	adantl 487	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) ∧ 𝑓 = (𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))) → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀))) ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
49	40, 48	rspcedv 3572	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 ((𝑠 ∈ 𝑆 ↦ if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀))) → ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
50	9, 15, 30, 49	mp3and 1493	. 2 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀))))
51	1, 12, 2, 26, 3	islindeps 49391	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
52	10, 11, 51	syl2anc 596	. 2 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
53	50, 52	mpbird 260	1 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088  Vcvv 3453  ifcif 4485  𝒫 cpw 4560   class class class wbr 5107   ↦ cmpt 5190  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830   finSupp cfsupp 9335  1c1 11129   < clt 11271  ♯chash 14398  Basecbs 17307  Scalarcsca 17351  0_gc0g 17530  1_rcur 20326  Ringcrg 20378  LModclmod 21050   linC clinc 49342   linDepS clindeps 49379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-n0 12533  df-xnn0 12606  df-z 12620  df-uz 12892  df-fz 13566  df-seq 14070  df-hash 14399  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-plusg 17361  df-0g 17532  df-gsum 17533  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-grp 19066  df-minusg 19067  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-lmod 21052  df-linc 49344  df-lininds 49380  df-lindeps 49382

This theorem is used by:  el0ldepsnzr  49405