Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smprngopr Structured version   Visualization version   GIF version

Theorem smprngopr 35322
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 1131 . 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 35295 . . . 4 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
543ad2ant1 1128 . . 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 35297 . . . . . . 7 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
10 eqcom 2826 . . . . . . 7 (𝑈 = 𝑍𝑍 = 𝑈)
11 eqcom 2826 . . . . . . 7 ({𝑍} = 𝑋𝑋 = {𝑍})
129, 10, 113bitr4g 316 . . . . . 6 (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋))
1312necon3bid 3058 . . . . 5 (𝑅 ∈ RingOps → (𝑈𝑍 ↔ {𝑍} ≠ 𝑋))
1413biimpa 479 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → {𝑍} ≠ 𝑋)
15143adant3 1127 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋)
16 df-pr 4562 . . . . . . . 8 {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋})
1716eqeq2i 2832 . . . . . . 7 ((Idl‘𝑅) = {{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}))
18 eleq2 2899 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋})))
19 eleq2 2899 . . . . . . . . 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 4123 . . . . . . . . . 10 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22 velsn 4575 . . . . . . . . . . 11 (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23 velsn 4575 . . . . . . . . . . 11 (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
2422, 23orbi12i 911 . . . . . . . . . 10 ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
2521, 24bitri 277 . . . . . . . . 9 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26 elun 4123 . . . . . . . . . 10 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27 velsn 4575 . . . . . . . . . . 11 (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28 velsn 4575 . . . . . . . . . . 11 (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
2927, 28orbi12i 911 . . . . . . . . . 10 ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3026, 29bitri 277 . . . . . . . . 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, 31syl6bb 289 . . . . . . 7 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
3317, 32sylbi 219 . . . . . 6 ((Idl‘𝑅) = {{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34333ad2ant3 1130 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
35 eqimss 4021 . . . . . . . . . . 11 (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍})
3635orcd 869 . . . . . . . . . 10 (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
3736adantr 483 . . . . . . . . 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 4021 . . . . . . . . . . 11 (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍})
4140olcd 870 . . . . . . . . . 10 (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
4241adantl 484 . . . . . . . . 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 483 . . . . . . . . 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 2842 . . . . . . . . . . . . 13 𝑋 = ran (1st𝑅)
5049, 6, 8rngo1cl 35209 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝑈𝑋)
5150adantr 483 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → 𝑈𝑋)
526, 49, 8rngolidm 35207 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑈𝐻𝑈) = 𝑈)
5350, 52mpdan 685 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈)
5453eleq1d 2895 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
558fvexi 6677 . . . . . . . . . . . . . . 15 𝑈 ∈ V
5655elsn 4574 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍)
5754, 56syl6bb 289 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍))
5857necon3bbid 3051 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (¬ (𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈𝑍))
5958biimpar 480 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍})
60 oveq1 7155 . . . . . . . . . . . . . 14 (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
6160eleq1d 2895 . . . . . . . . . . . . 13 (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍}))
6261notbid 320 . . . . . . . . . . . 12 (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍}))
63 oveq2 7156 . . . . . . . . . . . . . 14 (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈))
6463eleq1d 2895 . . . . . . . . . . . . 13 (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍}))
6564notbid 320 . . . . . . . . . . . 12 (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍}))
6662, 65rspc2ev 3633 . . . . . . . . . . 11 ((𝑈𝑋𝑈𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
6751, 51, 59, 66syl3anc 1366 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
68 rexnal2 3256 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
6967, 68sylib 220 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
7069pm2.21d 121 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
71 raleq 3404 . . . . . . . . . 10 (𝑖 = 𝑋 → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍}))
72 raleq 3404 . . . . . . . . . . 11 (𝑗 = 𝑋 → (∀𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7372ralbidv 3195 . . . . . . . . . 10 (𝑗 = 𝑋 → (∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7471, 73sylan9bb 512 . . . . . . . . 9 ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7574imbi1d 344 . . . . . . . 8 ((𝑖 = 𝑋𝑗 = 𝑋) → ((∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) ↔ (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7670, 75syl5ibrcom 249 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7739, 44, 47, 76ccased 1033 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
78773adant3 1127 . . . . 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 242 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
8079ralrimivv 3188 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
812, 6, 7ispridl 35304 . . . 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 1128 . . 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 1337 . 2 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅))
842, 3isprrngo 35320 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
851, 83, 84sylanbrc 585 1 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1082   = wceq 1531  wcel 2108  wne 3014  wral 3136  wrex 3137  cun 3932  wss 3934  {csn 4559  {cpr 4561  ran crn 5549  cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  GIdcgi 28259  RingOpscrngo 35164  Idlcidl 35277  PrIdlcpridl 35278  PrRingcprrng 35316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7106  df-ov 7151  df-1st 7681  df-2nd 7682  df-grpo 28262  df-gid 28263  df-ginv 28264  df-ablo 28314  df-ass 35113  df-exid 35115  df-mgmOLD 35119  df-sgrOLD 35131  df-mndo 35137  df-rngo 35165  df-idl 35280  df-pridl 35281  df-prrngo 35318
This theorem is referenced by:  divrngpr  35323
  Copyright terms: Public domain W3C validator