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Theorem smprngopr 37223
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 = (1st β€˜π‘…)
smprngpr.2 𝐻 = (2nd β€˜π‘…)
smprngpr.3 𝑋 = ran 𝐺
smprngpr.4 𝑍 = (GIdβ€˜πΊ)
smprngpr.5 π‘ˆ = (GIdβ€˜π»)
Assertion
Ref Expression
smprngopr ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ 𝑅 ∈ PrRing)
Proof of Theorem smprngopr
Dummy variables 𝑖 𝑗 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1134 . 2 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ 𝑅 ∈ RingOps)
2 smprngpr.1 . . . . 5 𝐺 = (1st β€˜π‘…)
3 smprngpr.4 . . . . 5 𝑍 = (GIdβ€˜πΊ)
42, 30idl 37196 . . . 4 (𝑅 ∈ RingOps β†’ {𝑍} ∈ (Idlβ€˜π‘…))
543ad2ant1 1131 . . 3 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ {𝑍} ∈ (Idlβ€˜π‘…))
6 smprngpr.2 . . . . . . . 8 𝐻 = (2nd β€˜π‘…)
7 smprngpr.3 . . . . . . . 8 𝑋 = ran 𝐺
8 smprngpr.5 . . . . . . . 8 π‘ˆ = (GIdβ€˜π»)
92, 6, 7, 3, 80rngo 37198 . . . . . . 7 (𝑅 ∈ RingOps β†’ (𝑍 = π‘ˆ ↔ 𝑋 = {𝑍}))
10 eqcom 2737 . . . . . . 7 (π‘ˆ = 𝑍 ↔ 𝑍 = π‘ˆ)
11 eqcom 2737 . . . . . . 7 ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
129, 10, 113bitr4g 313 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘ˆ = 𝑍 ↔ {𝑍} = 𝑋))
1312necon3bid 2983 . . . . 5 (𝑅 ∈ RingOps β†’ (π‘ˆ β‰  𝑍 ↔ {𝑍} β‰  𝑋))
1413biimpa 475 . . . 4 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ {𝑍} β‰  𝑋)
15143adant3 1130 . . 3 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ {𝑍} β‰  𝑋)
16 df-pr 4630 . . . . . . . 8 {{𝑍}, 𝑋} = ({{𝑍}} βˆͺ {𝑋})
1716eqeq2i 2743 . . . . . . 7 ((Idlβ€˜π‘…) = {{𝑍}, 𝑋} ↔ (Idlβ€˜π‘…) = ({{𝑍}} βˆͺ {𝑋}))
18 eleq2 2820 . . . . . . . . 9 ((Idlβ€˜π‘…) = ({{𝑍}} βˆͺ {𝑋}) β†’ (𝑖 ∈ (Idlβ€˜π‘…) ↔ 𝑖 ∈ ({{𝑍}} βˆͺ {𝑋})))
19 eleq2 2820 . . . . . . . . 9 ((Idlβ€˜π‘…) = ({{𝑍}} βˆͺ {𝑋}) β†’ (𝑗 ∈ (Idlβ€˜π‘…) ↔ 𝑗 ∈ ({{𝑍}} βˆͺ {𝑋})))
2018, 19anbi12d 629 . . . . . . . 8 ((Idlβ€˜π‘…) = ({{𝑍}} βˆͺ {𝑋}) β†’ ((𝑖 ∈ (Idlβ€˜π‘…) ∧ 𝑗 ∈ (Idlβ€˜π‘…)) ↔ (𝑖 ∈ ({{𝑍}} βˆͺ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} βˆͺ {𝑋}))))
21 elun 4147 . . . . . . . . . 10 (𝑖 ∈ ({{𝑍}} βˆͺ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22 velsn 4643 . . . . . . . . . . 11 (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23 velsn 4643 . . . . . . . . . . 11 (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
2422, 23orbi12i 911 . . . . . . . . . 10 ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
2521, 24bitri 274 . . . . . . . . 9 (𝑖 ∈ ({{𝑍}} βˆͺ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26 elun 4147 . . . . . . . . . 10 (𝑗 ∈ ({{𝑍}} βˆͺ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27 velsn 4643 . . . . . . . . . . 11 (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28 velsn 4643 . . . . . . . . . . 11 (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
2927, 28orbi12i 911 . . . . . . . . . 10 ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3026, 29bitri 274 . . . . . . . . 9 (𝑗 ∈ ({{𝑍}} βˆͺ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
3125, 30anbi12i 625 . . . . . . . 8 ((𝑖 ∈ ({{𝑍}} βˆͺ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} βˆͺ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))
3220, 31bitrdi 286 . . . . . . 7 ((Idlβ€˜π‘…) = ({{𝑍}} βˆͺ {𝑋}) β†’ ((𝑖 ∈ (Idlβ€˜π‘…) ∧ 𝑗 ∈ (Idlβ€˜π‘…)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
3317, 32sylbi 216 . . . . . 6 ((Idlβ€˜π‘…) = {{𝑍}, 𝑋} β†’ ((𝑖 ∈ (Idlβ€˜π‘…) ∧ 𝑗 ∈ (Idlβ€˜π‘…)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34333ad2ant3 1133 . . . . 5 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ ((𝑖 ∈ (Idlβ€˜π‘…) ∧ 𝑗 ∈ (Idlβ€˜π‘…)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
35 eqimss 4039 . . . . . . . . . . 11 (𝑖 = {𝑍} β†’ 𝑖 βŠ† {𝑍})
3635orcd 869 . . . . . . . . . 10 (𝑖 = {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))
3736adantr 479 . . . . . . . . 9 ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))
3837a1d 25 . . . . . . . 8 ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))
3938a1i 11 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
40 eqimss 4039 . . . . . . . . . . 11 (𝑗 = {𝑍} β†’ 𝑗 βŠ† {𝑍})
4140olcd 870 . . . . . . . . . 10 (𝑗 = {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))
4241adantl 480 . . . . . . . . 9 ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))
4342a1d 25 . . . . . . . 8 ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))
4443a1i 11 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
4536adantr 479 . . . . . . . . 9 ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))
4645a1d 25 . . . . . . . 8 ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))
4746a1i 11 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
482rneqi 5935 . . . . . . . . . . . . . 14 ran 𝐺 = ran (1st β€˜π‘…)
497, 48eqtri 2758 . . . . . . . . . . . . 13 𝑋 = ran (1st β€˜π‘…)
5049, 6, 8rngo1cl 37110 . . . . . . . . . . . 12 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
5150adantr 479 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ π‘ˆ ∈ 𝑋)
526, 49, 8rngolidm 37108 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘ˆπ»π‘ˆ) = π‘ˆ)
5350, 52mpdan 683 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps β†’ (π‘ˆπ»π‘ˆ) = π‘ˆ)
5453eleq1d 2816 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps β†’ ((π‘ˆπ»π‘ˆ) ∈ {𝑍} ↔ π‘ˆ ∈ {𝑍}))
558fvexi 6904 . . . . . . . . . . . . . . 15 π‘ˆ ∈ V
5655elsn 4642 . . . . . . . . . . . . . 14 (π‘ˆ ∈ {𝑍} ↔ π‘ˆ = 𝑍)
5754, 56bitrdi 286 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps β†’ ((π‘ˆπ»π‘ˆ) ∈ {𝑍} ↔ π‘ˆ = 𝑍))
5857necon3bbid 2976 . . . . . . . . . . . 12 (𝑅 ∈ RingOps β†’ (Β¬ (π‘ˆπ»π‘ˆ) ∈ {𝑍} ↔ π‘ˆ β‰  𝑍))
5958biimpar 476 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ Β¬ (π‘ˆπ»π‘ˆ) ∈ {𝑍})
60 oveq1 7418 . . . . . . . . . . . . . 14 (π‘₯ = π‘ˆ β†’ (π‘₯𝐻𝑦) = (π‘ˆπ»π‘¦))
6160eleq1d 2816 . . . . . . . . . . . . 13 (π‘₯ = π‘ˆ β†’ ((π‘₯𝐻𝑦) ∈ {𝑍} ↔ (π‘ˆπ»π‘¦) ∈ {𝑍}))
6261notbid 317 . . . . . . . . . . . 12 (π‘₯ = π‘ˆ β†’ (Β¬ (π‘₯𝐻𝑦) ∈ {𝑍} ↔ Β¬ (π‘ˆπ»π‘¦) ∈ {𝑍}))
63 oveq2 7419 . . . . . . . . . . . . . 14 (𝑦 = π‘ˆ β†’ (π‘ˆπ»π‘¦) = (π‘ˆπ»π‘ˆ))
6463eleq1d 2816 . . . . . . . . . . . . 13 (𝑦 = π‘ˆ β†’ ((π‘ˆπ»π‘¦) ∈ {𝑍} ↔ (π‘ˆπ»π‘ˆ) ∈ {𝑍}))
6564notbid 317 . . . . . . . . . . . 12 (𝑦 = π‘ˆ β†’ (Β¬ (π‘ˆπ»π‘¦) ∈ {𝑍} ↔ Β¬ (π‘ˆπ»π‘ˆ) ∈ {𝑍}))
6662, 65rspc2ev 3623 . . . . . . . . . . 11 ((π‘ˆ ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋 ∧ Β¬ (π‘ˆπ»π‘ˆ) ∈ {𝑍}) β†’ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝑋 Β¬ (π‘₯𝐻𝑦) ∈ {𝑍})
6751, 51, 59, 66syl3anc 1369 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝑋 Β¬ (π‘₯𝐻𝑦) ∈ {𝑍})
68 rexnal2 3133 . . . . . . . . . 10 (βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝑋 Β¬ (π‘₯𝐻𝑦) ∈ {𝑍} ↔ Β¬ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍})
6967, 68sylib 217 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ Β¬ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍})
7069pm2.21d 121 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))
71 raleq 3320 . . . . . . . . . 10 (𝑖 = 𝑋 β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍}))
72 raleq 3320 . . . . . . . . . . 11 (𝑗 = 𝑋 β†’ (βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} ↔ βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍}))
7372ralbidv 3175 . . . . . . . . . 10 (𝑗 = 𝑋 β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍}))
7471, 73sylan9bb 508 . . . . . . . . 9 ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍}))
7574imbi1d 340 . . . . . . . 8 ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) β†’ ((βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})) ↔ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
7670, 75syl5ibrcom 246 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
7739, 44, 47, 76ccased 1035 . . . . . 6 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍) β†’ (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
78773adant3 1130 . . . . 5 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
7934, 78sylbid 239 . . . 4 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ ((𝑖 ∈ (Idlβ€˜π‘…) ∧ 𝑗 ∈ (Idlβ€˜π‘…)) β†’ (βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍}))))
8079ralrimivv 3196 . . 3 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ βˆ€π‘– ∈ (Idlβ€˜π‘…)βˆ€π‘— ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))
812, 6, 7ispridl 37205 . . . 4 (𝑅 ∈ RingOps β†’ ({𝑍} ∈ (PrIdlβ€˜π‘…) ↔ ({𝑍} ∈ (Idlβ€˜π‘…) ∧ {𝑍} β‰  𝑋 ∧ βˆ€π‘– ∈ (Idlβ€˜π‘…)βˆ€π‘— ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))))
82813ad2ant1 1131 . . 3 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ ({𝑍} ∈ (PrIdlβ€˜π‘…) ↔ ({𝑍} ∈ (Idlβ€˜π‘…) ∧ {𝑍} β‰  𝑋 ∧ βˆ€π‘– ∈ (Idlβ€˜π‘…)βˆ€π‘— ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ 𝑖 βˆ€π‘¦ ∈ 𝑗 (π‘₯𝐻𝑦) ∈ {𝑍} β†’ (𝑖 βŠ† {𝑍} ∨ 𝑗 βŠ† {𝑍})))))
835, 15, 80, 82mpbir3and 1340 . 2 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ {𝑍} ∈ (PrIdlβ€˜π‘…))
842, 3isprrngo 37221 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)))
851, 83, 84sylanbrc 581 1 ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ 𝑅 ∈ PrRing)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068   βˆͺ cun 3945   βŠ† wss 3947  {csn 4627  {cpr 4629  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  GIdcgi 30010  RingOpscrngo 37065  Idlcidl 37178  PrIdlcpridl 37179  PrRingcprrng 37217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-1st 7977  df-2nd 7978  df-grpo 30013  df-gid 30014  df-ginv 30015  df-ablo 30065  df-ass 37014  df-exid 37016  df-mgmOLD 37020  df-sgrOLD 37032  df-mndo 37038  df-rngo 37066  df-idl 37181  df-pridl 37182  df-prrngo 37219
This theorem is referenced by:  divrngpr  37224
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