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

Theorem el0ldep 42773
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 2770 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2770 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 eqid 2770 . . . . 5 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4 eqid 2770 . . . . 5 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5 eqeq1 2774 . . . . . . 7 (𝑠 = 𝑦 → (𝑠 = (0g𝑀) ↔ 𝑦 = (0g𝑀)))
65ifbid 4245 . . . . . 6 (𝑠 = 𝑦 → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
76cbvmptv 4882 . . . . 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 42679 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
983adant1r 1186 . . 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 1238 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11 simp2 1130 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12 eqid 2770 . . . . 5 (0g𝑀) = (0g𝑀)
13 eqid 2770 . . . . 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 42730 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀))
1510, 11, 14syl2anc 565 . . 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 1131 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (0g𝑀) ∈ 𝑆)
17 fveq2 6332 . . . . . 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 3001 . . . . 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 467 . . . 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 fvexd 6344 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
21 iftrue 4229 . . . . . . 7 (𝑠 = (0g𝑀) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
2221, 13fvmptg 6422 . . . . . 6 (((0g𝑀) ∈ 𝑆 ∧ (1r‘(Scalar‘𝑀)) ∈ V) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)) = (1r‘(Scalar‘𝑀)))
2316, 20, 22syl2anc 565 . . . . 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)))
242lmodring 19080 . . . . . . . 8 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
2524anim1i 594 . . . . . . 7 ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
26253ad2ant1 1126 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))))
27 eqid 2770 . . . . . . 7 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2827, 4, 3ring1ne0 18798 . . . . . 6 (((Scalar‘𝑀) ∈ Ring ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
2926, 28syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
3023, 29eqnetrd 3009 . . . 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)))
3116, 19, 30rspcedvd 3465 . . 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)))
322, 27, 4lmod1cl 19099 . . . . . . . . . 10 (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
332, 27, 3lmod0cl 19098 . . . . . . . . . 10 (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
3432, 33ifcld 4268 . . . . . . . . 9 (𝑀 ∈ LMod → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
3534adantr 466 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
36353ad2ant1 1126 . . . . . . 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)))
3736adantr 466 . . . . . 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)))
3837, 13fmptd 6527 . . . . 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)))
39 fvexd 6344 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
4039, 11elmapd 8022 . . . . 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))))
4138, 40mpbird 247 . . . 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))
42 breq1 4787 . . . . . 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))))
43 oveq1 6799 . . . . . . 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))
4443eqeq1d 2772 . . . . . 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)))
45 fveq1 6331 . . . . . . . 8 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓𝑥) = ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥))
4645neeq1d 3001 . . . . . . 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))))
4746rexbidv 3199 . . . . . 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))))
4842, 44, 473anbi123d 1546 . . . . 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)))))
4948adantl 467 . . . 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)))))
5041, 49rspcedv 3462 . . 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)))))
519, 15, 31, 50mp3and 1574 . 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))))
521, 12, 2, 27, 3islindeps 42760 . . 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)))))
5310, 11, 52syl2anc 565 . 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)))))
5451, 53mpbird 247 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 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wrex 3061  Vcvv 3349  ifcif 4223  𝒫 cpw 4295   class class class wbr 4784  cmpt 4861  wf 6027  cfv 6031  (class class class)co 6792  𝑚 cmap 8008   finSupp cfsupp 8430  1c1 10138   < clt 10275  chash 13320  Basecbs 16063  Scalarcsca 16151  0gc0g 16307  1rcur 18708  Ringcrg 18754  LModclmod 19072   linC clinc 42711   linDepS clindeps 42748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-fz 12533  df-seq 13008  df-hash 13321  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-plusg 16161  df-0g 16309  df-gsum 16310  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-mgp 18697  df-ur 18709  df-ring 18756  df-lmod 19074  df-linc 42713  df-lininds 42749  df-lindeps 42751
This theorem is referenced by:  el0ldepsnzr  42774
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