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Theorem el0ldep 41543
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 2621 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2621 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 eqid 2621 . . . . 5 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
4 eqid 2621 . . . . 5 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
5 eqeq1 2625 . . . . . . 7 (𝑠 = 𝑦 → (𝑠 = (0g𝑀) ↔ 𝑦 = (0g𝑀)))
65ifbid 4080 . . . . . 6 (𝑠 = 𝑦 → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
76cbvmptv 4710 . . . . 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 41449 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
983adant1r 1316 . . 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 1083 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑀 ∈ LMod)
11 simp2 1060 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 ∈ 𝒫 (Base‘𝑀))
12 eqid 2621 . . . . 5 (0g𝑀) = (0g𝑀)
13 eqid 2621 . . . . 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 41500 . . . 4 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑆) = (0g𝑀))
1510, 11, 14syl2anc 692 . . 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 1061 . . . 4 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (0g𝑀) ∈ 𝑆)
17 fveq2 6148 . . . . . 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 2849 . . . . 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 482 . . . 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 fvex 6158 . . . . . . 7 (1r‘(Scalar‘𝑀)) ∈ V
2120a1i 11 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ∈ V)
22 iftrue 4064 . . . . . . 7 (𝑠 = (0g𝑀) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = (1r‘(Scalar‘𝑀)))
2322, 13fvmptg 6237 . . . . . 6 (((0g𝑀) ∈ 𝑆 ∧ (1r‘(Scalar‘𝑀)) ∈ V) → ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘(0g𝑀)) = (1r‘(Scalar‘𝑀)))
2416, 21, 23syl2anc 692 . . . . 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)))
252lmodring 18792 . . . . . . . 8 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
2625anim1i 591 . . . . . . 7 ((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))))
27263ad2ant1 1080 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → ((Scalar‘𝑀) ∈ Ring ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))))
28 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2928, 4, 3ring1ne0 18512 . . . . . 6 (((Scalar‘𝑀) ∈ Ring ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
3027, 29syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (1r‘(Scalar‘𝑀)) ≠ (0g‘(Scalar‘𝑀)))
3124, 30eqnetrd 2857 . . . 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)))
3216, 19, 31rspcedvd 3302 . . 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)))
332, 28, 4lmod1cl 18811 . . . . . . . . . 10 (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
342, 28, 3lmod0cl 18810 . . . . . . . . . 10 (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
3533, 34ifcld 4103 . . . . . . . . 9 (𝑀 ∈ LMod → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
3635adantr 481 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) → if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
37363ad2ant1 1080 . . . . . . 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)))
3837adantr 481 . . . . . 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)))
3938, 13fmptd 6340 . . . . 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)))
40 fvex 6158 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
4140a1i 11 . . . . . 6 (((𝑀 ∈ LMod ∧ 1 < (#‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → (Base‘(Scalar‘𝑀)) ∈ V)
4241, 11elmapd 7816 . . . . 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))))
4339, 42mpbird 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))
44 breq1 4616 . . . . . 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))))
45 oveq1 6611 . . . . . . 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))
4645eqeq1d 2623 . . . . . 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)))
47 fveq1 6147 . . . . . . . 8 (𝑓 = (𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓𝑥) = ((𝑠𝑆 ↦ if(𝑠 = (0g𝑀), (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))‘𝑥))
4847neeq1d 2849 . . . . . . 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))))
4948rexbidv 3045 . . . . . 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))))
5044, 46, 493anbi123d 1396 . . . . 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)))))
5150adantl 482 . . . 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)))))
5243, 51rspcedv 3299 . . 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)))))
539, 15, 32, 52mp3and 1424 . 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))))
541, 12, 2, 28, 3islindeps 41530 . . 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)))))
5510, 11, 54syl2anc 692 . 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)))))
5653, 55mpbird 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  ifcif 4058  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802   finSupp cfsupp 8219  1c1 9881   < clt 10018  #chash 13057  Basecbs 15781  Scalarcsca 15865  0gc0g 16021  1rcur 18422  Ringcrg 18468  LModclmod 18784   linC clinc 41481   linDepS clindeps 41518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-seq 12742  df-hash 13058  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-linc 41483  df-lininds 41519  df-lindeps 41521
This theorem is referenced by:  el0ldepsnzr  41544
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