Theorem el0ldep 48195

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 2740	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2740	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2740	. . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4		eqid 2740	. . . . 5 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5		eqeq1 2744	. . . . . . 7 ⊢ (𝑠 = 𝑦 → (𝑠 = (0g‘𝑀) ↔ 𝑦 = (0g‘𝑀)))
6	5	ifbid 4571	. . . . . 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 5279	. . . . 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 48105	. . . 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 1177	. . 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 1197	. . . 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 2740	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
13		eqid 2740	. . . . 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 48152	. . . 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 583	. . 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 6920	. . . . . 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 3006	. . . . 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 4554	. . . . . 6 ⊢ (𝑠 = (0g‘𝑀) → if(𝑠 = (0g‘𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21		fvexd 6935	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
22	13, 20, 16, 21	fvmptd3 7052	. . . . 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 20888	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
24	23	anim1i 614	. . . . . . 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 2740	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27	26, 4, 3	ring1ne0 20322	. . . . . 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 3014	. . . 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 3637	. . 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 20909	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
32	2, 26, 3	lmod0cl 20908	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
33	31, 32	ifcld 4594	. . . . . . . . 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 7149	. . . . 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 6935	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
39	38, 11	elmapd 8898	. . . . 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 5169	. . . . . 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 7455	. . . . . . 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 2742	. . . . . 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 6919	. . . . . . . 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 3006	. . . . . . 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 3185	. . . . . 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 1436	. . . . 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 3628	. . 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 1464	. 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 48182	. . 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 583	. 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 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  Vcvv 3488  ifcif 4548  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884   finSupp cfsupp 9431  1c1 11185   < clt 11324  ♯chash 14379  Basecbs 17258  Scalarcsca 17314  0_gc0g 17499  1_rcur 20208  Ringcrg 20260  LModclmod 20880   linC clinc 48133   linDepS clindeps 48170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053  df-hash 14380  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-lmod 20882  df-linc 48135  df-lininds 48171  df-lindeps 48173

This theorem is referenced by:  el0ldepsnzr  48196