Theorem el0ldep 48954

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 2737	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2737	. . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4		eqid 2737	. . . . 5 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5		eqeq1 2741	. . . . . . 7 ⊢ (𝑠 = 𝑦 → (𝑠 = (0g‘𝑀) ↔ 𝑦 = (0g‘𝑀)))
6	5	ifbid 4491	. . . . . 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 5190	. . . . 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 48865	. . . 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 1179	. . 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 1199	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11		simp2 1138	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12		eqid 2737	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
13		eqid 2737	. . . . 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 48911	. . . 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 585	. . 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 1139	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (0g‘𝑀) ∈ 𝑆)
17		fveq2 6834	. . . . . 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 2992	. . . . 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 481	. . . 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 4473	. . . . . 6 ⊢ (𝑠 = (0g‘𝑀) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21		fvexd 6849	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
22	13, 20, 16, 21	fvmptd3 6965	. . . . 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 20854	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
24	23	anim1i 616	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
25	24	3ad2ant1 1134	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
26		eqid 2737	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27	26, 4, 3	ring1ne0 20271	. . . . . 6 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
28	25, 27	syl 17	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
29	22, 28	eqnetrd 3000	. . . 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 3567	. . 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 20875	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
32	2, 26, 3	lmod0cl 20874	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
33	31, 32	ifcld 4514	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
35	34	3ad2ant1 1134	. . . . . . 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 480	. . . . . 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 7061	. . . . 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 6849	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
39	38, 11	elmapd 8780	. . . . 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 257	. . . 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 5089	. . . . . 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 7367	. . . . . . 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 2739	. . . . . 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 6833	. . . . . . . 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 2992	. . . . . . 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 3162	. . . . . 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 1439	. . . . 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 481	. . . 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 3558	. . 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 1467	. 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 48941	. . 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 585	. 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 257	1 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  Vcvv 3430  ifcif 4467  𝒫 cpw 4542   class class class wbr 5086   ↦ cmpt 5167  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766   finSupp cfsupp 9267  1c1 11030   < clt 11170  ♯chash 14283  Basecbs 17170  Scalarcsca 17214  0_gc0g 17393  1_rcur 20153  Ringcrg 20205  LModclmod 20846   linC clinc 48892   linDepS clindeps 48929

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-fz 13453  df-seq 13955  df-hash 14284  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-0g 17395  df-gsum 17396  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-lmod 20848  df-linc 48894  df-lininds 48930  df-lindeps 48932

This theorem is referenced by:  el0ldepsnzr  48955