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Theorem lindsrng01 48313
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 20888), 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 20888 . . . . . . . 8 (𝑀 ∈ LMod → 𝐸 ≠ ∅)
42fvexi 6920 . . . . . . . . . 10 𝐸 ∈ V
5 hasheq0 14398 . . . . . . . . . 10 (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
64, 5ax-mp 5 . . . . . . . . 9 ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7 eqneqall 2948 . . . . . . . . . 10 (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
87com12 32 . . . . . . . . 9 (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
96, 8biimtrid 242 . . . . . . . 8 (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
103, 9syl 17 . . . . . . 7 (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
1110adantr 480 . . . . . 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 20882 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑅 ∈ Ring)
1413adantr 480 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15 eqid 2734 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
162, 150ring 20542 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
1714, 16sylan 580 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
18 simpr 484 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
1918adantr 480 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
2019adantl 481 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21 snex 5441 . . . . . . . . . . . . . 14 {(0g𝑅)} ∈ V
2219, 21jctil 519 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
2322adantl 481 . . . . . . . . . . . 12 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24 elmapg 8877 . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . 14 (0g𝑅) ∈ V
2726fconst2 7224 . . . . . . . . . . . . 13 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑆 × {(0g𝑅)}))
28 fconstmpt 5750 . . . . . . . . . . . . . 14 (𝑆 × {(0g𝑅)}) = (𝑥𝑆 ↦ (0g𝑅))
2928eqeq2i 2747 . . . . . . . . . . . . 13 (𝑓 = (𝑆 × {(0g𝑅)}) ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
3027, 29bitri 275 . . . . . . . . . . . 12 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
31 eqidd 2735 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (𝑥𝑆 ↦ (0g𝑅)) = (𝑥𝑆 ↦ (0g𝑅)))
32 eqidd 2735 . . . . . . . . . . . . . . . 16 ((((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) ∧ 𝑥 = 𝑣) → (0g𝑅) = (0g𝑅))
33 simpr 484 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → 𝑣𝑆)
34 fvexd 6921 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (0g𝑅) ∈ V)
3531, 32, 33, 34fvmptd 7022 . . . . . . . . . . . . . . 15 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))
3635ralrimiva 3143 . . . . . . . . . . . . . 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 5150 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓 finSupp (0g𝑅) ↔ (𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅)))
39 oveq1 7437 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆))
4039eqeq1d 2736 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓( linC ‘𝑀)𝑆) = (0g𝑀) ↔ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀)))
4138, 40anbi12d 632 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓 finSupp (0g𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) ↔ ((𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅) ∧ ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆) = (0g𝑀))))
42 fveq1 6905 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓𝑣) = ((𝑥𝑆 ↦ (0g𝑅))‘𝑣))
4342eqeq1d 2736 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓𝑣) = (0g𝑅) ↔ ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4443ralbidv 3175 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (∀𝑣𝑆 (𝑓𝑣) = (0g𝑅) ↔ ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4541, 44imbi12d 344 . . . . . . . . . . . . 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 247 . . . . . . . . . . . 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 242 . . . . . . . . . . 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 240 . . . . . . . . . 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 3142 . . . . . . . . 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 7437 . . . . . . . . . . 11 (𝐸 = {(0g𝑅)} → (𝐸m 𝑆) = ({(0g𝑅)} ↑m 𝑆))
5150raleqdv 3323 . . . . . . . . . 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 480 . . . . . . . . 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 257 . . . . . . . 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 482 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
5554ancomd 461 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
5655adantl 481 . . . . . . . . 9 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
57 lindsrng01.b . . . . . . . . . 10 𝐵 = (Base‘𝑀)
58 eqid 2734 . . . . . . . . . 10 (0g𝑀) = (0g𝑀)
5957, 58, 1, 2, 15islininds 48291 . . . . . . . . 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 713 . . . . . . 7 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
6217, 61mpancom 688 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
6362expcom 413 . . . . 5 ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6412, 63jaoi 857 . . . 4 (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6564expd 415 . . 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 1110 1 ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864   finSupp cfsupp 9398  0cc0 11152  1c1 11153  chash 14365  Basecbs 17244  Scalarcsca 17300  0gc0g 17485  Ringcrg 20250  LModclmod 20874   linC clinc 48249   linIndS clininds 48285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-ring 20252  df-lmod 20876  df-lininds 48287
This theorem is referenced by:  lindszr  48314
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