Theorem el0ldep 48459

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 2730	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2730	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2730	. . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4		eqid 2730	. . . . 5 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5		eqeq1 2734	. . . . . . 7 ⊢ (𝑠 = 𝑦 → (𝑠 = (0g‘𝑀) ↔ 𝑦 = (0g‘𝑀)))
6	5	ifbid 4515	. . . . . 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 5214	. . . . 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 48369	. . . 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 1178	. . 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 1198	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11		simp2 1137	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12		eqid 2730	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
13		eqid 2730	. . . . 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 48416	. . . 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 584	. . 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 1138	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (0g‘𝑀) ∈ 𝑆)
17		fveq2 6861	. . . . . 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 2985	. . . . 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 4497	. . . . . 6 ⊢ (𝑠 = (0g‘𝑀) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21		fvexd 6876	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
22	13, 20, 16, 21	fvmptd3 6994	. . . . 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 20781	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
24	23	anim1i 615	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
25	24	3ad2ant1 1133	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
26		eqid 2730	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27	26, 4, 3	ring1ne0 20215	. . . . . 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 2993	. . . 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 3593	. . 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 20802	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
32	2, 26, 3	lmod0cl 20801	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
33	31, 32	ifcld 4538	. . . . . . . . 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 1133	. . . . . . 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 7090	. . . . 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 6876	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
39	38, 11	elmapd 8816	. . . . 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 5113	. . . . . 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 7397	. . . . . . 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 2732	. . . . . 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 6860	. . . . . . . 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 2985	. . . . . . 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 3158	. . . . . 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 1438	. . . . 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 3584	. . 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 1466	. 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 48446	. . 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 584	. 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 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  Vcvv 3450  ifcif 4491  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802   finSupp cfsupp 9319  1c1 11076   < clt 11215  ♯chash 14302  Basecbs 17186  Scalarcsca 17230  0_gc0g 17409  1_rcur 20097  Ringcrg 20149  LModclmod 20773   linC clinc 48397   linDepS clindeps 48434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-seq 13974  df-hash 14303  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-0g 17411  df-gsum 17412  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-lmod 20775  df-linc 48399  df-lininds 48435  df-lindeps 48437

This theorem is referenced by:  el0ldepsnzr  48460