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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  el0ldep Structured version   Visualization version   GIF version

Theorem el0ldep 47335
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) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Proof of Theorem el0ldep
Dummy variables 𝑓 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2724 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 eqid 2724 . . . . 5 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4 eqid 2724 . . . . 5 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5 eqeq1 2728 . . . . . . 7 (𝑠 = 𝑦 → (𝑠 = (0g𝑀) ↔ 𝑦 = (0g𝑀)))
65ifbid 4543 . . . . . 6 (𝑠 = 𝑦 → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
76cbvmptv 5251 . . . . 5 (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) = (𝑦𝑆 ↦ if(𝑦 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
81, 2, 3, 4, 7mptcfsupp 47245 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
983adant1r 1174 . . 3 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
10 simp1l 1194 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11 simp2 1134 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12 eqid 2724 . . . . 5 (0g𝑀) = (0g𝑀)
13 eqid 2724 . . . . 5 (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
141, 2, 3, 4, 12, 13linc0scn0 47292 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀))
1510, 11, 14syl2anc 583 . . 3 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀))
16 simp3 1135 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (0g𝑀) ∈ 𝑆)
17 fveq2 6881 . . . . . 6 (𝑥 = (0g𝑀) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) = ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)))
1817neeq1d 2992 . . . . 5 (𝑥 = (0g𝑀) → (((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)) ≠ (0g‘(Scalar‘𝑀))))
1918adantl 481 . . . 4 ((((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) ∧ 𝑥 = (0g𝑀)) → (((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)) ≠ (0g‘(Scalar‘𝑀))))
20 iftrue 4526 . . . . . 6 (𝑠 = (0g𝑀) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21 fvexd 6896 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
2213, 20, 16, 21fvmptd3 7011 . . . . 5 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)) = (1r‘(Scalar‘𝑀)))
232lmodring 20704 . . . . . . . 8 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
2423anim1i 614 . . . . . . 7 ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
25243ad2ant1 1130 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
26 eqid 2724 . . . . . . 7 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2726, 4, 3ring1ne0 20188 . . . . . 6 (((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
2825, 27syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
2922, 28eqnetrd 3000 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)) ≠ (0g‘(Scalar‘𝑀)))
3016, 19, 29rspcedvd 3606 . . 3 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ∃𝑥𝑆 ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)))
312, 26, 4lmod1cl 20725 . . . . . . . . . 10 (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
322, 26, 3lmod0cl 20724 . . . . . . . . . 10 (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
3331, 32ifcld 4566 . . . . . . . . 9 (𝑀 ∈ LMod → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
3433adantr 480 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
35343ad2ant1 1130 . . . . . . 7 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
3635adantr 480 . . . . . 6 ((((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) ∧ 𝑠𝑆) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
3736fmpttd 7106 . . . . 5 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑆⟶(Base‘(Scalar‘𝑀)))
38 fvexd 6896 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
3938, 11elmapd 8830 . . . . 5 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆) ↔ (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑆⟶(Base‘(Scalar‘𝑀))))
4037, 39mpbird 257 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆))
41 breq1 5141 . . . . . 6 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀))))
42 oveq1 7408 . . . . . . 7 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆))
4342eqeq1d 2726 . . . . . 6 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ↔ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀)))
44 fveq1 6880 . . . . . . . 8 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓𝑥) = ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥))
4544neeq1d 2992 . . . . . . 7 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀))))
4645rexbidv 3170 . . . . . 6 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀)) ↔ ∃𝑥𝑆 ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀))))
4741, 43, 463anbi123d 1432 . . . . 5 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀))) ↔ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
4847adantl 481 . . . 4 ((((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) ∧ 𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))) → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀))) ↔ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥) ≠ (0g‘(Scalar‘𝑀)))))
4940, 48rspcedv 3597 . . 3 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (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) = (0g𝑀) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀)))))
509, 15, 30, 49mp3and 1460 . 2 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀))))
511, 12, 2, 26, 3islindeps 47322 . . 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) = (0g𝑀) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀)))))
5210, 11, 51syl2anc 583 . 2 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ (0g‘(Scalar‘𝑀)))))
5350, 52mpbird 257 1 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wrex 3062  Vcvv 3466  ifcif 4520  𝒫 cpw 4594   class class class wbr 5138  cmpt 5221  wf 6529  cfv 6533  (class class class)co 7401  m cmap 8816   finSupp cfsupp 9357  1c1 11107   < clt 11245  chash 14287  Basecbs 17143  Scalarcsca 17199  0gc0g 17384  1rcur 20076  Ringcrg 20128  LModclmod 20696   linC clinc 47273   linDepS clindeps 47310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-fz 13482  df-seq 13964  df-hash 14288  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-0g 17386  df-gsum 17387  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18856  df-minusg 18857  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-lmod 20698  df-linc 47275  df-lininds 47311  df-lindeps 47313
This theorem is referenced by:  el0ldepsnzr  47336
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