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Mirrors > Home > MPE Home > Th. List > Mathboxes > divrngpr | Structured version Visualization version GIF version |
Description: A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
divrngpr | ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2821 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2821 | . . . 4 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
4 | eqid 2821 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
5 | 1, 2, 3, 4 | isdrngo1 35233 | . . 3 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd ‘𝑅) ↾ ((ran (1st ‘𝑅) ∖ {(GId‘(1st ‘𝑅))}) × (ran (1st ‘𝑅) ∖ {(GId‘(1st ‘𝑅))}))) ∈ GrpOp)) |
6 | 5 | simplbi 500 | . 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
7 | eqid 2821 | . . 3 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
8 | 1, 2, 4, 3, 7 | dvrunz 35231 | . 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2nd ‘𝑅)) ≠ (GId‘(1st ‘𝑅))) |
9 | 1, 2, 4, 3 | divrngidl 35305 | . 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st ‘𝑅))}, ran (1st ‘𝑅)}) |
10 | 1, 2, 4, 3, 7 | smprngopr 35329 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2nd ‘𝑅)) ≠ (GId‘(1st ‘𝑅)) ∧ (Idl‘𝑅) = {{(GId‘(1st ‘𝑅))}, ran (1st ‘𝑅)}) → 𝑅 ∈ PrRing) |
11 | 6, 8, 9, 10 | syl3anc 1367 | 1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 {csn 4566 {cpr 4568 × cxp 5552 ran crn 5555 ↾ cres 5556 ‘cfv 6354 1st c1st 7686 2nd c2nd 7687 GrpOpcgr 28265 GIdcgi 28266 RingOpscrngo 35171 DivRingOpscdrng 35225 Idlcidl 35284 PrRingcprrng 35323 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-om 7580 df-1st 7688 df-2nd 7689 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-grpo 28269 df-gid 28270 df-ginv 28271 df-ablo 28321 df-ass 35120 df-exid 35122 df-mgmOLD 35126 df-sgrOLD 35138 df-mndo 35144 df-rngo 35172 df-drngo 35226 df-idl 35287 df-pridl 35288 df-prrngo 35325 |
This theorem is referenced by: flddmn 35335 |
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