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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 2823 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2823 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2823 | . . . 4 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
4 | eqid 2823 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
5 | 1, 2, 3, 4 | isdrngo1 35236 | . . 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 2823 | . . 3 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
8 | 1, 2, 4, 3, 7 | dvrunz 35234 | . 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2nd ‘𝑅)) ≠ (GId‘(1st ‘𝑅))) |
9 | 1, 2, 4, 3 | divrngidl 35308 | . 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st ‘𝑅))}, ran (1st ‘𝑅)}) |
10 | 1, 2, 4, 3, 7 | smprngopr 35332 | . 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 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 {cpr 4571 × cxp 5555 ran crn 5558 ↾ cres 5559 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 GrpOpcgr 28268 GIdcgi 28269 RingOpscrngo 35174 DivRingOpscdrng 35228 Idlcidl 35287 PrRingcprrng 35326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-om 7583 df-1st 7691 df-2nd 7692 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-grpo 28272 df-gid 28273 df-ginv 28274 df-ablo 28324 df-ass 35123 df-exid 35125 df-mgmOLD 35129 df-sgrOLD 35141 df-mndo 35147 df-rngo 35175 df-drngo 35229 df-idl 35290 df-pridl 35291 df-prrngo 35328 |
This theorem is referenced by: flddmn 35338 |
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