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Theorem smprngopr 38039
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 1135 . 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 38012 . . . 4 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
543ad2ant1 1132 . . 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 38014 . . . . . . 7 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
10 eqcom 2742 . . . . . . 7 (𝑈 = 𝑍𝑍 = 𝑈)
11 eqcom 2742 . . . . . . 7 ({𝑍} = 𝑋𝑋 = {𝑍})
129, 10, 113bitr4g 314 . . . . . 6 (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋))
1312necon3bid 2983 . . . . 5 (𝑅 ∈ RingOps → (𝑈𝑍 ↔ {𝑍} ≠ 𝑋))
1413biimpa 476 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → {𝑍} ≠ 𝑋)
15143adant3 1131 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋)
16 df-pr 4634 . . . . . . . 8 {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋})
1716eqeq2i 2748 . . . . . . 7 ((Idl‘𝑅) = {{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}))
18 eleq2 2828 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋})))
19 eleq2 2828 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑗 ∈ (Idl‘𝑅) ↔ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})))
2018, 19anbi12d 632 . . . . . . . 8 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋}))))
21 elun 4163 . . . . . . . . . 10 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22 velsn 4647 . . . . . . . . . . 11 (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23 velsn 4647 . . . . . . . . . . 11 (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
2422, 23orbi12i 914 . . . . . . . . . 10 ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
2521, 24bitri 275 . . . . . . . . 9 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26 elun 4163 . . . . . . . . . 10 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27 velsn 4647 . . . . . . . . . . 11 (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28 velsn 4647 . . . . . . . . . . 11 (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
2927, 28orbi12i 914 . . . . . . . . . 10 ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3026, 29bitri 275 . . . . . . . . 9 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3125, 30anbi12i 628 . . . . . . . 8 ((𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))
3220, 31bitrdi 287 . . . . . . 7 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
3317, 32sylbi 217 . . . . . 6 ((Idl‘𝑅) = {{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34333ad2ant3 1134 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
35 eqimss 4054 . . . . . . . . . . 11 (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍})
3635orcd 873 . . . . . . . . . 10 (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
3736adantr 480 . . . . . . . . 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 4054 . . . . . . . . . . 11 (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍})
4140olcd 874 . . . . . . . . . 10 (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
4241adantl 481 . . . . . . . . 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 480 . . . . . . . . 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 5951 . . . . . . . . . . . . . 14 ran 𝐺 = ran (1st𝑅)
497, 48eqtri 2763 . . . . . . . . . . . . 13 𝑋 = ran (1st𝑅)
5049, 6, 8rngo1cl 37926 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝑈𝑋)
5150adantr 480 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → 𝑈𝑋)
526, 49, 8rngolidm 37924 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑈𝐻𝑈) = 𝑈)
5350, 52mpdan 687 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈)
5453eleq1d 2824 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
558fvexi 6921 . . . . . . . . . . . . . . 15 𝑈 ∈ V
5655elsn 4646 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍)
5754, 56bitrdi 287 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍))
5857necon3bbid 2976 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (¬ (𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈𝑍))
5958biimpar 477 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍})
60 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
6160eleq1d 2824 . . . . . . . . . . . . 13 (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍}))
6261notbid 318 . . . . . . . . . . . 12 (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍}))
63 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈))
6463eleq1d 2824 . . . . . . . . . . . . 13 (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍}))
6564notbid 318 . . . . . . . . . . . 12 (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍}))
6662, 65rspc2ev 3635 . . . . . . . . . . 11 ((𝑈𝑋𝑈𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
6751, 51, 59, 66syl3anc 1370 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
68 rexnal2 3133 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
6967, 68sylib 218 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
7069pm2.21d 121 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
71 raleq 3321 . . . . . . . . . 10 (𝑖 = 𝑋 → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍}))
72 raleq 3321 . . . . . . . . . . 11 (𝑗 = 𝑋 → (∀𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7372ralbidv 3176 . . . . . . . . . 10 (𝑗 = 𝑋 → (∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7471, 73sylan9bb 509 . . . . . . . . 9 ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7574imbi1d 341 . . . . . . . 8 ((𝑖 = 𝑋𝑗 = 𝑋) → ((∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) ↔ (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7670, 75syl5ibrcom 247 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7739, 44, 47, 76ccased 1038 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
78773adant3 1131 . . . . 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 240 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
8079ralrimivv 3198 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
812, 6, 7ispridl 38021 . . . 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 1132 . . 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 1341 . 2 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅))
842, 3isprrngo 38037 . 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 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cun 3961  wss 3963  {csn 4631  {cpr 4633  ran crn 5690  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  GIdcgi 30519  RingOpscrngo 37881  Idlcidl 37994  PrIdlcpridl 37995  PrRingcprrng 38033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854  df-rngo 37882  df-idl 37997  df-pridl 37998  df-prrngo 38035
This theorem is referenced by:  divrngpr  38040
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