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Theorem smprngopr 35211
Description: A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
smprngpr.1 𝐺 = (1st𝑅)
smprngpr.2 𝐻 = (2nd𝑅)
smprngpr.3 𝑋 = ran 𝐺
smprngpr.4 𝑍 = (GId‘𝐺)
smprngpr.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
smprngopr ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Proof of Theorem smprngopr
Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1128 . 2 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ RingOps)
2 smprngpr.1 . . . . 5 𝐺 = (1st𝑅)
3 smprngpr.4 . . . . 5 𝑍 = (GId‘𝐺)
42, 30idl 35184 . . . 4 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
543ad2ant1 1125 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (Idl‘𝑅))
6 smprngpr.2 . . . . . . . 8 𝐻 = (2nd𝑅)
7 smprngpr.3 . . . . . . . 8 𝑋 = ran 𝐺
8 smprngpr.5 . . . . . . . 8 𝑈 = (GId‘𝐻)
92, 6, 7, 3, 80rngo 35186 . . . . . . 7 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
10 eqcom 2825 . . . . . . 7 (𝑈 = 𝑍𝑍 = 𝑈)
11 eqcom 2825 . . . . . . 7 ({𝑍} = 𝑋𝑋 = {𝑍})
129, 10, 113bitr4g 315 . . . . . 6 (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋))
1312necon3bid 3057 . . . . 5 (𝑅 ∈ RingOps → (𝑈𝑍 ↔ {𝑍} ≠ 𝑋))
1413biimpa 477 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → {𝑍} ≠ 𝑋)
15143adant3 1124 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋)
16 df-pr 4560 . . . . . . . 8 {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋})
1716eqeq2i 2831 . . . . . . 7 ((Idl‘𝑅) = {{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}))
18 eleq2 2898 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋})))
19 eleq2 2898 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑗 ∈ (Idl‘𝑅) ↔ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})))
2018, 19anbi12d 630 . . . . . . . 8 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋}))))
21 elun 4122 . . . . . . . . . 10 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22 velsn 4573 . . . . . . . . . . 11 (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23 velsn 4573 . . . . . . . . . . 11 (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
2422, 23orbi12i 908 . . . . . . . . . 10 ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
2521, 24bitri 276 . . . . . . . . 9 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26 elun 4122 . . . . . . . . . 10 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27 velsn 4573 . . . . . . . . . . 11 (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28 velsn 4573 . . . . . . . . . . 11 (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
2927, 28orbi12i 908 . . . . . . . . . 10 ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3026, 29bitri 276 . . . . . . . . 9 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3125, 30anbi12i 626 . . . . . . . 8 ((𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))
3220, 31syl6bb 288 . . . . . . 7 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
3317, 32sylbi 218 . . . . . 6 ((Idl‘𝑅) = {{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34333ad2ant3 1127 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
35 eqimss 4020 . . . . . . . . . . 11 (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍})
3635orcd 869 . . . . . . . . . 10 (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
3736adantr 481 . . . . . . . . 9 ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
3837a1d 25 . . . . . . . 8 ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
3938a1i 11 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
40 eqimss 4020 . . . . . . . . . . 11 (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍})
4140olcd 870 . . . . . . . . . 10 (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
4241adantl 482 . . . . . . . . 9 ((𝑖 = 𝑋𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
4342a1d 25 . . . . . . . 8 ((𝑖 = 𝑋𝑗 = {𝑍}) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
4443a1i 11 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = 𝑋𝑗 = {𝑍}) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
4536adantr 481 . . . . . . . . 9 ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
4645a1d 25 . . . . . . . 8 ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
4746a1i 11 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
482rneqi 5800 . . . . . . . . . . . . . 14 ran 𝐺 = ran (1st𝑅)
497, 48eqtri 2841 . . . . . . . . . . . . 13 𝑋 = ran (1st𝑅)
5049, 6, 8rngo1cl 35098 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝑈𝑋)
5150adantr 481 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → 𝑈𝑋)
526, 49, 8rngolidm 35096 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑈𝐻𝑈) = 𝑈)
5350, 52mpdan 683 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈)
5453eleq1d 2894 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
558fvexi 6677 . . . . . . . . . . . . . . 15 𝑈 ∈ V
5655elsn 4572 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍)
5754, 56syl6bb 288 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍))
5857necon3bbid 3050 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (¬ (𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈𝑍))
5958biimpar 478 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍})
60 oveq1 7152 . . . . . . . . . . . . . 14 (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
6160eleq1d 2894 . . . . . . . . . . . . 13 (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍}))
6261notbid 319 . . . . . . . . . . . 12 (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍}))
63 oveq2 7153 . . . . . . . . . . . . . 14 (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈))
6463eleq1d 2894 . . . . . . . . . . . . 13 (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍}))
6564notbid 319 . . . . . . . . . . . 12 (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍}))
6662, 65rspc2ev 3632 . . . . . . . . . . 11 ((𝑈𝑋𝑈𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
6751, 51, 59, 66syl3anc 1363 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
68 rexnal2 3255 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
6967, 68sylib 219 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
7069pm2.21d 121 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
71 raleq 3403 . . . . . . . . . 10 (𝑖 = 𝑋 → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍}))
72 raleq 3403 . . . . . . . . . . 11 (𝑗 = 𝑋 → (∀𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7372ralbidv 3194 . . . . . . . . . 10 (𝑗 = 𝑋 → (∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7471, 73sylan9bb 510 . . . . . . . . 9 ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7574imbi1d 343 . . . . . . . 8 ((𝑖 = 𝑋𝑗 = 𝑋) → ((∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) ↔ (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7670, 75syl5ibrcom 248 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7739, 44, 47, 76ccased 1030 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
78773adant3 1124 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7934, 78sylbid 241 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
8079ralrimivv 3187 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
812, 6, 7ispridl 35193 . . . 4 (𝑅 ∈ RingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))))
82813ad2ant1 1125 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))))
835, 15, 80, 82mpbir3and 1334 . 2 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅))
842, 3isprrngo 35209 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
851, 83, 84sylanbrc 583 1 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  cun 3931  wss 3933  {csn 4557  {cpr 4559  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  GIdcgi 28194  RingOpscrngo 35053  Idlcidl 35166  PrIdlcpridl 35167  PrRingcprrng 35205
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 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gid 28198  df-ginv 28199  df-ablo 28249  df-ass 35002  df-exid 35004  df-mgmOLD 35008  df-sgrOLD 35020  df-mndo 35026  df-rngo 35054  df-idl 35169  df-pridl 35170  df-prrngo 35207
This theorem is referenced by:  divrngpr  35212
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