Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lindsrng01 Structured version   Visualization version   GIF version

Theorem lindsrng01 47788
Description: Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 20797), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
Hypotheses
Ref Expression
lindsrng01.b 𝐵 = (Base‘𝑀)
lindsrng01.r 𝑅 = (Scalar‘𝑀)
lindsrng01.e 𝐸 = (Base‘𝑅)
Assertion
Ref Expression
lindsrng01 ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Proof of Theorem lindsrng01
Dummy variables 𝑓 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lindsrng01.r . . . . . . . . 9 𝑅 = (Scalar‘𝑀)
2 lindsrng01.e . . . . . . . . 9 𝐸 = (Base‘𝑅)
31, 2lmodsn0 20797 . . . . . . . 8 (𝑀 ∈ LMod → 𝐸 ≠ ∅)
42fvexi 6914 . . . . . . . . . 10 𝐸 ∈ V
5 hasheq0 14375 . . . . . . . . . 10 (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
64, 5ax-mp 5 . . . . . . . . 9 ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7 eqneqall 2940 . . . . . . . . . 10 (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
87com12 32 . . . . . . . . 9 (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
96, 8biimtrid 241 . . . . . . . 8 (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
103, 9syl 17 . . . . . . 7 (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
1110adantr 479 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
1211com12 32 . . . . 5 ((♯‘𝐸) = 0 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
131lmodring 20791 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑅 ∈ Ring)
1413adantr 479 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15 eqid 2725 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
162, 150ring 20503 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
1714, 16sylan 578 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
18 simpr 483 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
1918adantr 479 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
2019adantl 480 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21 snex 5436 . . . . . . . . . . . . . 14 {(0g𝑅)} ∈ V
2219, 21jctil 518 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
2322adantl 480 . . . . . . . . . . . 12 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24 elmapg 8867 . . . . . . . . . . . 12 (({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑓 ∈ ({(0g𝑅)} ↑m 𝑆) ↔ 𝑓:𝑆⟶{(0g𝑅)}))
2523, 24syl 17 . . . . . . . . . . 11 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0g𝑅)} ↑m 𝑆) ↔ 𝑓:𝑆⟶{(0g𝑅)}))
26 fvex 6913 . . . . . . . . . . . . . 14 (0g𝑅) ∈ V
2726fconst2 7221 . . . . . . . . . . . . 13 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑆 × {(0g𝑅)}))
28 fconstmpt 5743 . . . . . . . . . . . . . 14 (𝑆 × {(0g𝑅)}) = (𝑥𝑆 ↦ (0g𝑅))
2928eqeq2i 2738 . . . . . . . . . . . . 13 (𝑓 = (𝑆 × {(0g𝑅)}) ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
3027, 29bitri 274 . . . . . . . . . . . 12 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
31 eqidd 2726 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (𝑥𝑆 ↦ (0g𝑅)) = (𝑥𝑆 ↦ (0g𝑅)))
32 eqidd 2726 . . . . . . . . . . . . . . . 16 ((((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) ∧ 𝑥 = 𝑣) → (0g𝑅) = (0g𝑅))
33 simpr 483 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → 𝑣𝑆)
34 fvexd 6915 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (0g𝑅) ∈ V)
3531, 32, 33, 34fvmptd 7015 . . . . . . . . . . . . . . 15 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))
3635ralrimiva 3135 . . . . . . . . . . . . . 14 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))
3736a1d 25 . . . . . . . . . . . . 13 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (((𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅) ∧ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
38 breq1 5155 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓 finSupp (0g𝑅) ↔ (𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅)))
39 oveq1 7430 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆))
4039eqeq1d 2727 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ↔ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀)))
4138, 40anbi12d 630 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) ↔ ((𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅) ∧ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀))))
42 fveq1 6899 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓𝑣) = ((𝑥𝑆 ↦ (0g𝑅))‘𝑣))
4342eqeq1d 2727 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓𝑣) = (0g𝑅) ↔ ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4443ralbidv 3167 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (∀𝑣𝑆 (𝑓𝑣) = (0g𝑅) ↔ ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4541, 44imbi12d 343 . . . . . . . . . . . . 13 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)) ↔ (((𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅) ∧ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))))
4637, 45syl5ibrcom 246 . . . . . . . . . . . 12 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
4730, 46biimtrid 241 . . . . . . . . . . 11 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓:𝑆⟶{(0g𝑅)} → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
4825, 47sylbid 239 . . . . . . . . . 10 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0g𝑅)} ↑m 𝑆) → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
4948ralrimiv 3134 . . . . . . . . 9 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ ({(0g𝑅)} ↑m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)))
50 oveq1 7430 . . . . . . . . . . 11 (𝐸 = {(0g𝑅)} → (𝐸m 𝑆) = ({(0g𝑅)} ↑m 𝑆))
5150raleqdv 3314 . . . . . . . . . 10 (𝐸 = {(0g𝑅)} → (∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)) ↔ ∀𝑓 ∈ ({(0g𝑅)} ↑m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
5251adantr 479 . . . . . . . . 9 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)) ↔ ∀𝑓 ∈ ({(0g𝑅)} ↑m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
5349, 52mpbird 256 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)))
54 simpl 481 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
5554ancomd 460 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
5655adantl 480 . . . . . . . . 9 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
57 lindsrng01.b . . . . . . . . . 10 𝐵 = (Base‘𝑀)
58 eqid 2725 . . . . . . . . . 10 (0g𝑀) = (0g𝑀)
5957, 58, 1, 2, 15islininds 47766 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)))))
6056, 59syl 17 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)))))
6120, 53, 60mpbir2and 711 . . . . . . 7 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
6217, 61mpancom 686 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
6362expcom 412 . . . . 5 ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6412, 63jaoi 855 . . . 4 (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6564expd 414 . . 3 (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑀 ∈ LMod → (𝑆 ∈ 𝒫 𝐵𝑆 linIndS 𝑀)))
6665com12 32 . 2 (𝑀 ∈ LMod → (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵𝑆 linIndS 𝑀)))
67663imp 1108 1 ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  c0 4324  𝒫 cpw 4606  {csn 4632   class class class wbr 5152  cmpt 5235   × cxp 5679  wf 6549  cfv 6553  (class class class)co 7423  m cmap 8854   finSupp cfsupp 9401  0cc0 11154  1c1 11155  chash 14342  Basecbs 17208  Scalarcsca 17264  0gc0g 17449  Ringcrg 20211  LModclmod 20783   linC clinc 47724   linIndS clininds 47760
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 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1st 8002  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8856  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 9978  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-nn 12260  df-n0 12520  df-z 12606  df-uz 12870  df-fz 13534  df-hash 14343  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-ring 20213  df-lmod 20785  df-lininds 47762
This theorem is referenced by:  lindszr  47789
  Copyright terms: Public domain W3C validator