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

Theorem el0ldep 44873
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 2798 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2798 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 eqid 2798 . . . . 5 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4 eqid 2798 . . . . 5 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5 eqeq1 2802 . . . . . . 7 (𝑠 = 𝑦 → (𝑠 = (0g𝑀) ↔ 𝑦 = (0g𝑀)))
65ifbid 4447 . . . . . 6 (𝑠 = 𝑦 → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
76cbvmptv 5133 . . . . 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 44780 . . . 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 2798 . . . . 5 (0g𝑀) = (0g𝑀)
13 eqid 2798 . . . . 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 44830 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀))
1510, 11, 14syl2anc 587 . . 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 6645 . . . . . 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 3046 . . . . 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 485 . . . 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 4431 . . . . . 6 (𝑠 = (0g𝑀) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21 fvexd 6660 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
2213, 20, 16, 21fvmptd3 6768 . . . . 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 19635 . . . . . . . 8 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
2423anim1i 617 . . . . . . 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 2798 . . . . . . 7 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2726, 4, 3ring1ne0 19337 . . . . . 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 3054 . . . 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 3574 . . 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 19654 . . . . . . . . . 10 (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
322, 26, 3lmod0cl 19653 . . . . . . . . . 10 (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
3331, 32ifcld 4470 . . . . . . . . 9 (𝑀 ∈ LMod → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
3433adantr 484 . . . . . . . 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 484 . . . . . 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 6856 . . . . 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 6660 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
3938, 11elmapd 8403 . . . . 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 260 . . . 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 5033 . . . . . 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 7142 . . . . . . 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 2800 . . . . . 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 6644 . . . . . . . 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 3046 . . . . . . 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 3256 . . . . . 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 1433 . . . . 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 485 . . . 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 3564 . . 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 1461 . 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 44860 . . 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 587 . 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 260 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  wrex 3107  Vcvv 3441  ifcif 4425  𝒫 cpw 4497   class class class wbr 5030  cmpt 5110  wf 6320  cfv 6324  (class class class)co 7135  m cmap 8389   finSupp cfsupp 8817  1c1 10527   < clt 10664  chash 13686  Basecbs 16475  Scalarcsca 16560  0gc0g 16705  1rcur 19244  Ringcrg 19290  LModclmod 19627   linC clinc 44811   linDepS clindeps 44848
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-seq 13365  df-hash 13687  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-mgp 19233  df-ur 19245  df-ring 19292  df-lmod 19629  df-linc 44813  df-lininds 44849  df-lindeps 44851
This theorem is referenced by:  el0ldepsnzr  44874
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