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Theorem lindsrng01 42785
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 19098), 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 19098 . . . . . . . 8 (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4 fvex 6363 . . . . . . . . . . 11 (Base‘𝑅) ∈ V
52, 4eqeltri 2835 . . . . . . . . . 10 𝐸 ∈ V
6 hasheq0 13366 . . . . . . . . . 10 (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
75, 6ax-mp 5 . . . . . . . . 9 ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
8 eqneqall 2943 . . . . . . . . . 10 (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
98com12 32 . . . . . . . . 9 (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
107, 9syl5bi 232 . . . . . . . 8 (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
113, 10syl 17 . . . . . . 7 (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
1211adantr 472 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
1312com12 32 . . . . 5 ((♯‘𝐸) = 0 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
141lmodring 19093 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑅 ∈ Ring)
1514adantr 472 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
16 eqid 2760 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
172, 160ring 19492 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
1815, 17sylan 489 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
19 simpr 479 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
2019adantr 472 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
2120adantl 473 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
22 snex 5057 . . . . . . . . . . . . . 14 {(0g𝑅)} ∈ V
2320, 22jctil 561 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
2423adantl 473 . . . . . . . . . . . 12 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
25 elmapg 8038 . . . . . . . . . . . 12 (({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑓 ∈ ({(0g𝑅)} ↑𝑚 𝑆) ↔ 𝑓:𝑆⟶{(0g𝑅)}))
2624, 25syl 17 . . . . . . . . . . 11 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0g𝑅)} ↑𝑚 𝑆) ↔ 𝑓:𝑆⟶{(0g𝑅)}))
27 fvex 6363 . . . . . . . . . . . . . 14 (0g𝑅) ∈ V
2827fconst2 6635 . . . . . . . . . . . . 13 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑆 × {(0g𝑅)}))
29 fconstmpt 5320 . . . . . . . . . . . . . 14 (𝑆 × {(0g𝑅)}) = (𝑥𝑆 ↦ (0g𝑅))
3029eqeq2i 2772 . . . . . . . . . . . . 13 (𝑓 = (𝑆 × {(0g𝑅)}) ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
3128, 30bitri 264 . . . . . . . . . . . 12 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
32 eqidd 2761 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (𝑥𝑆 ↦ (0g𝑅)) = (𝑥𝑆 ↦ (0g𝑅)))
33 eqidd 2761 . . . . . . . . . . . . . . . 16 ((((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) ∧ 𝑥 = 𝑣) → (0g𝑅) = (0g𝑅))
34 simpr 479 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → 𝑣𝑆)
35 fvexd 6365 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (0g𝑅) ∈ V)
3632, 33, 34, 35fvmptd 6451 . . . . . . . . . . . . . . 15 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))
3736ralrimiva 3104 . . . . . . . . . . . . . 14 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))
3837a1d 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)))
39 breq1 4807 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓 finSupp (0g𝑅) ↔ (𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅)))
40 oveq1 6821 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆))
4140eqeq1d 2762 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ↔ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀)))
4239, 41anbi12d 749 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) ↔ ((𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅) ∧ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀))))
43 fveq1 6352 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓𝑣) = ((𝑥𝑆 ↦ (0g𝑅))‘𝑣))
4443eqeq1d 2762 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓𝑣) = (0g𝑅) ↔ ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4544ralbidv 3124 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (∀𝑣𝑆 (𝑓𝑣) = (0g𝑅) ↔ ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4642, 45imbi12d 333 . . . . . . . . . . . . 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))))
4738, 46syl5ibrcom 237 . . . . . . . . . . . 12 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
4831, 47syl5bi 232 . . . . . . . . . . 11 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓:𝑆⟶{(0g𝑅)} → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅))))
4926, 48sylbid 230 . . . . . . . . . 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))))
5049ralrimiv 3103 . . . . . . . . 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)))
51 oveq1 6821 . . . . . . . . . . 11 (𝐸 = {(0g𝑅)} → (𝐸𝑚 𝑆) = ({(0g𝑅)} ↑𝑚 𝑆))
5251raleqdv 3283 . . . . . . . . . 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))))
5352adantr 472 . . . . . . . . 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))))
5450, 53mpbird 247 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑣𝑆 (𝑓𝑣) = (0g𝑅)))
55 simpl 474 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
5655ancomd 466 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
5756adantl 473 . . . . . . . . 9 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
58 lindsrng01.b . . . . . . . . . 10 𝐵 = (Base‘𝑀)
59 eqid 2760 . . . . . . . . . 10 (0g𝑀) = (0g𝑀)
6058, 59, 1, 2, 16islininds 42763 . . . . . . . . 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)))))
6157, 60syl 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)))))
6221, 54, 61mpbir2and 995 . . . . . . 7 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
6318, 62mpancom 706 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
6463expcom 450 . . . . 5 ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6513, 64jaoi 393 . . . 4 (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6665expd 451 . . 3 (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑀 ∈ LMod → (𝑆 ∈ 𝒫 𝐵𝑆 linIndS 𝑀)))
6766com12 32 . 2 (𝑀 ∈ LMod → (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵𝑆 linIndS 𝑀)))
68673imp 1102 1 ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  Vcvv 3340  c0 4058  𝒫 cpw 4302  {csn 4321   class class class wbr 4804  cmpt 4881   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6814  𝑚 cmap 8025   finSupp cfsupp 8442  0cc0 10148  1c1 10149  chash 13331  Basecbs 16079  Scalarcsca 16166  0gc0g 16322  Ringcrg 18767  LModclmod 19085   linC clinc 42721   linIndS clininds 42757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-ring 18769  df-lmod 19087  df-lininds 42759
This theorem is referenced by:  lindszr  42786
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