Theorem el0ldep 48942

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 2736	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2736	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2736	. . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4		eqid 2736	. . . . 5 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5		eqeq1 2740	. . . . . . 7 ⊢ (𝑠 = 𝑦 → (𝑠 = (0g‘𝑀) ↔ 𝑦 = (0g‘𝑀)))
6	5	ifbid 4490	. . . . . 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 5189	. . . . 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 48853	. . . 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 2736	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
13		eqid 2736	. . . . 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 48899	. . . 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 6840	. . . . . 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 2991	. . . . 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 4472	. . . . . 6 ⊢ (𝑠 = (0g‘𝑀) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21		fvexd 6855	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
22	13, 20, 16, 21	fvmptd3 6971	. . . . 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 20863	. . . . . . . 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 2736	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27	26, 4, 3	ring1ne0 20280	. . . . . 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 2999	. . . 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 3566	. . 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 20884	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
32	2, 26, 3	lmod0cl 20883	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
33	31, 32	ifcld 4513	. . . . . . . . 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 7067	. . . . 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 6855	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
39	38, 11	elmapd 8787	. . . . 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 5088	. . . . . 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 7374	. . . . . . 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 2738	. . . . . 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 6839	. . . . . . . 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 2991	. . . . . . 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 3161	. . . . . 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 3557	. . 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 48929	. . 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 2932  ∃wrex 3061  Vcvv 3429  ifcif 4466  𝒫 cpw 4541   class class class wbr 5085   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   finSupp cfsupp 9274  1c1 11039   < clt 11179  ♯chash 14292  Basecbs 17179  Scalarcsca 17223  0_gc0g 17402  1_rcur 20162  Ringcrg 20214  LModclmod 20855   linC clinc 48880   linDepS clindeps 48917

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-seq 13964  df-hash 14293  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-0g 17404  df-gsum 17405  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-lmod 20857  df-linc 48882  df-lininds 48918  df-lindeps 48920

This theorem is referenced by:  el0ldepsnzr  48943