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 34183
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 1130 . 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 34156 . . . 4 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
543ad2ant1 1127 . . 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 34158 . . . . . . 7 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
10 eqcom 2778 . . . . . . 7 (𝑈 = 𝑍𝑍 = 𝑈)
11 eqcom 2778 . . . . . . 7 ({𝑍} = 𝑋𝑋 = {𝑍})
129, 10, 113bitr4g 303 . . . . . 6 (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋))
1312necon3bid 2987 . . . . 5 (𝑅 ∈ RingOps → (𝑈𝑍 ↔ {𝑍} ≠ 𝑋))
1413biimpa 462 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → {𝑍} ≠ 𝑋)
15143adant3 1126 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋)
16 df-pr 4319 . . . . . . . 8 {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋})
1716eqeq2i 2783 . . . . . . 7 ((Idl‘𝑅) = {{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}))
18 eleq2 2839 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋})))
19 eleq2 2839 . . . . . . . . 9 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑗 ∈ (Idl‘𝑅) ↔ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})))
2018, 19anbi12d 616 . . . . . . . 8 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋}))))
21 elun 3904 . . . . . . . . . 10 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22 velsn 4332 . . . . . . . . . . 11 (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23 velsn 4332 . . . . . . . . . . 11 (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
2422, 23orbi12i 900 . . . . . . . . . 10 ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
2521, 24bitri 264 . . . . . . . . 9 (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26 elun 3904 . . . . . . . . . 10 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27 velsn 4332 . . . . . . . . . . 11 (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28 velsn 4332 . . . . . . . . . . 11 (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
2927, 28orbi12i 900 . . . . . . . . . 10 ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3026, 29bitri 264 . . . . . . . . 9 (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3125, 30anbi12i 612 . . . . . . . 8 ((𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))
3220, 31syl6bb 276 . . . . . . 7 ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
3317, 32sylbi 207 . . . . . 6 ((Idl‘𝑅) = {{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34333ad2ant3 1129 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
35 eqimss 3806 . . . . . . . . . . 11 (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍})
3635orcd 862 . . . . . . . . . 10 (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
3736adantr 466 . . . . . . . . 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 3806 . . . . . . . . . . 11 (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍})
4140olcd 863 . . . . . . . . . 10 (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
4241adantl 467 . . . . . . . . 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 466 . . . . . . . . 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 5490 . . . . . . . . . . . . . 14 ran 𝐺 = ran (1st𝑅)
497, 48eqtri 2793 . . . . . . . . . . . . 13 𝑋 = ran (1st𝑅)
5049, 6, 8rngo1cl 34070 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝑈𝑋)
5150adantr 466 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → 𝑈𝑋)
526, 49, 8rngolidm 34068 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑈𝐻𝑈) = 𝑈)
5350, 52mpdan 667 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈)
5453eleq1d 2835 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
55 fvex 6342 . . . . . . . . . . . . . . . 16 (GId‘𝐻) ∈ V
568, 55eqeltri 2846 . . . . . . . . . . . . . . 15 𝑈 ∈ V
5756elsn 4331 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍)
5854, 57syl6bb 276 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍))
5958necon3bbid 2980 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (¬ (𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈𝑍))
6059biimpar 463 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍})
61 oveq1 6800 . . . . . . . . . . . . . 14 (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
6261eleq1d 2835 . . . . . . . . . . . . 13 (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍}))
6362notbid 307 . . . . . . . . . . . 12 (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍}))
64 oveq2 6801 . . . . . . . . . . . . . 14 (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈))
6564eleq1d 2835 . . . . . . . . . . . . 13 (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍}))
6665notbid 307 . . . . . . . . . . . 12 (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍}))
6763, 66rspc2ev 3474 . . . . . . . . . . 11 ((𝑈𝑋𝑈𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
6851, 51, 60, 67syl3anc 1476 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
69 rexnal2 3191 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
7068, 69sylib 208 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ¬ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
7170pm2.21d 119 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
72 raleq 3287 . . . . . . . . . 10 (𝑖 = 𝑋 → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍}))
73 raleq 3287 . . . . . . . . . . 11 (𝑗 = 𝑋 → (∀𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7473ralbidv 3135 . . . . . . . . . 10 (𝑗 = 𝑋 → (∀𝑥𝑋𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7572, 74sylan9bb 499 . . . . . . . . 9 ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
7675imbi1d 330 . . . . . . . 8 ((𝑖 = 𝑋𝑗 = 𝑋) → ((∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) ↔ (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7771, 76syl5ibrcom 237 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → ((𝑖 = 𝑋𝑗 = 𝑋) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
7839, 44, 47, 77ccased 1023 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
79783adant3 1126 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
8034, 79sylbid 230 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) → (∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
8180ralrimivv 3119 . . 3 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
822, 6, 7ispridl 34165 . . . 4 (𝑅 ∈ RingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥𝑖𝑦𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))))
83823ad2ant1 1127 . . 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})))))
845, 15, 81, 83mpbir3and 1427 . 2 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅))
852, 3isprrngo 34181 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
861, 84, 85sylanbrc 572 1 ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cun 3721  wss 3723  {csn 4316  {cpr 4318  ran crn 5250  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  GIdcgi 27684  RingOpscrngo 34025  Idlcidl 34138  PrIdlcpridl 34139  PrRingcprrng 34177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-1st 7315  df-2nd 7316  df-grpo 27687  df-gid 27688  df-ginv 27689  df-ablo 27739  df-ass 33974  df-exid 33976  df-mgmOLD 33980  df-sgrOLD 33992  df-mndo 33998  df-rngo 34026  df-idl 34141  df-pridl 34142  df-prrngo 34179
This theorem is referenced by:  divrngpr  34184
  Copyright terms: Public domain W3C validator