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Theorem lindsrng01 44421
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 19578), 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 19578 . . . . . . . 8 (𝑀 ∈ LMod → 𝐸 ≠ ∅)
42fvexi 6678 . . . . . . . . . 10 𝐸 ∈ V
5 hasheq0 13714 . . . . . . . . . 10 (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
64, 5ax-mp 5 . . . . . . . . 9 ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7 eqneqall 3027 . . . . . . . . . 10 (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
87com12 32 . . . . . . . . 9 (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
96, 8syl5bi 243 . . . . . . . 8 (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
103, 9syl 17 . . . . . . 7 (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
1110adantr 481 . . . . . 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 19573 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑅 ∈ Ring)
1413adantr 481 . . . . . . . 8 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15 eqid 2821 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
162, 150ring 19973 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
1714, 16sylan 580 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g𝑅)})
18 simpr 485 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
1918adantr 481 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
2019adantl 482 . . . . . . . 8 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21 snex 5323 . . . . . . . . . . . . . 14 {(0g𝑅)} ∈ V
2219, 21jctil 520 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
2322adantl 482 . . . . . . . . . . . 12 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24 elmapg 8409 . . . . . . . . . . . 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 6677 . . . . . . . . . . . . . 14 (0g𝑅) ∈ V
2726fconst2 6960 . . . . . . . . . . . . 13 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑆 × {(0g𝑅)}))
28 fconstmpt 5608 . . . . . . . . . . . . . 14 (𝑆 × {(0g𝑅)}) = (𝑥𝑆 ↦ (0g𝑅))
2928eqeq2i 2834 . . . . . . . . . . . . 13 (𝑓 = (𝑆 × {(0g𝑅)}) ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
3027, 29bitri 276 . . . . . . . . . . . 12 (𝑓:𝑆⟶{(0g𝑅)} ↔ 𝑓 = (𝑥𝑆 ↦ (0g𝑅)))
31 eqidd 2822 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (𝑥𝑆 ↦ (0g𝑅)) = (𝑥𝑆 ↦ (0g𝑅)))
32 eqidd 2822 . . . . . . . . . . . . . . . 16 ((((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) ∧ 𝑥 = 𝑣) → (0g𝑅) = (0g𝑅))
33 simpr 485 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → 𝑣𝑆)
34 fvexd 6679 . . . . . . . . . . . . . . . 16 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → (0g𝑅) ∈ V)
3531, 32, 33, 34fvmptd 6768 . . . . . . . . . . . . . . 15 (((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣𝑆) → ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅))
3635ralrimiva 3182 . . . . . . . . . . . . . 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 5061 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓 finSupp (0g𝑅) ↔ (𝑥𝑆 ↦ (0g𝑅)) finSupp (0g𝑅)))
39 oveq1 7152 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥𝑆 ↦ (0g𝑅))( linC ‘𝑀)𝑆))
4039eqeq1d 2823 . . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (𝑓𝑣) = ((𝑥𝑆 ↦ (0g𝑅))‘𝑣))
4342eqeq1d 2823 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → ((𝑓𝑣) = (0g𝑅) ↔ ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4443ralbidv 3197 . . . . . . . . . . . . . 14 (𝑓 = (𝑥𝑆 ↦ (0g𝑅)) → (∀𝑣𝑆 (𝑓𝑣) = (0g𝑅) ↔ ∀𝑣𝑆 ((𝑥𝑆 ↦ (0g𝑅))‘𝑣) = (0g𝑅)))
4541, 44imbi12d 346 . . . . . . . . . . . . 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 248 . . . . . . . . . . . 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, 46syl5bi 243 . . . . . . . . . . 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 241 . . . . . . . . . 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 3181 . . . . . . . . 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 7152 . . . . . . . . . . 11 (𝐸 = {(0g𝑅)} → (𝐸m 𝑆) = ({(0g𝑅)} ↑m 𝑆))
5150raleqdv 3416 . . . . . . . . . 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 481 . . . . . . . . 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 258 . . . . . . . 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 483 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
5554ancomd 462 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
5655adantl 482 . . . . . . . . 9 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod))
57 lindsrng01.b . . . . . . . . . 10 𝐵 = (Base‘𝑀)
58 eqid 2821 . . . . . . . . . 10 (0g𝑀) = (0g𝑀)
5957, 58, 1, 2, 15islininds 44399 . . . . . . . . 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 709 . . . . . . 7 ((𝐸 = {(0g𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
6217, 61mpancom 684 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
6362expcom 414 . . . . 5 ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6412, 63jaoi 851 . . . 4 (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
6564expd 416 . . 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 1103 1 ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3016  wral 3138  Vcvv 3495  c0 4290  𝒫 cpw 4537  {csn 4559   class class class wbr 5058  cmpt 5138   × cxp 5547  wf 6345  cfv 6349  (class class class)co 7145  m cmap 8396   finSupp cfsupp 8822  0cc0 10526  1c1 10527  chash 13680  Basecbs 16473  Scalarcsca 16558  0gc0g 16703  Ringcrg 19228  LModclmod 19565   linC clinc 44357   linIndS clininds 44393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681  df-0g 16705  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-grp 18046  df-ring 19230  df-lmod 19567  df-lininds 44395
This theorem is referenced by:  lindszr  44422
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