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

Theorem el0ldep 48312
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 2735 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2735 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 eqid 2735 . . . . 5 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4 eqid 2735 . . . . 5 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5 eqeq1 2739 . . . . . . 7 (𝑠 = 𝑦 → (𝑠 = (0g𝑀) ↔ 𝑦 = (0g𝑀)))
65ifbid 4554 . . . . . 6 (𝑠 = 𝑦 → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
76cbvmptv 5261 . . . . 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 48222 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
983adant1r 1176 . . 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 1196 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11 simp2 1136 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12 eqid 2735 . . . . 5 (0g𝑀) = (0g𝑀)
13 eqid 2735 . . . . 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 48269 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀))
1510, 11, 14syl2anc 584 . . 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 1137 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (0g𝑀) ∈ 𝑆)
17 fveq2 6907 . . . . . 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 2998 . . . . 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 4537 . . . . . 6 (𝑠 = (0g𝑀) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
21 fvexd 6922 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
2213, 20, 16, 21fvmptd3 7039 . . . . 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 20883 . . . . . . . 8 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
2423anim1i 615 . . . . . . 7 ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
25243ad2ant1 1132 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
26 eqid 2735 . . . . . . 7 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2726, 4, 3ring1ne0 20313 . . . . . 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 3006 . . . 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 3624 . . 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 20904 . . . . . . . . . 10 (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
322, 26, 3lmod0cl 20903 . . . . . . . . . 10 (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
3331, 32ifcld 4577 . . . . . . . . 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 1132 . . . . . . 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 7135 . . . . 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 6922 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
3938, 11elmapd 8879 . . . . 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 5151 . . . . . 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 7438 . . . . . . 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 2737 . . . . . 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 6906 . . . . . . . 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 2998 . . . . . . 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 3177 . . . . . 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 1435 . . . . 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 3615 . . 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 1463 . 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 48299 . . 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 584 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  ifcif 4531  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865   finSupp cfsupp 9399  1c1 11154   < clt 11293  chash 14366  Basecbs 17245  Scalarcsca 17301  0gc0g 17486  1rcur 20199  Ringcrg 20251  LModclmod 20875   linC clinc 48250   linDepS clindeps 48287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-hash 14367  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-lmod 20877  df-linc 48252  df-lininds 48288  df-lindeps 48290
This theorem is referenced by:  el0ldepsnzr  48313
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