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Mathbox for Jeff Madsen |
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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 2733 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2733 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2733 | . . . 4 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
4 | eqid 2733 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
5 | 1, 2, 3, 4 | isdrngo1 36824 | . . 3 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd ‘𝑅) ↾ ((ran (1st ‘𝑅) ∖ {(GId‘(1st ‘𝑅))}) × (ran (1st ‘𝑅) ∖ {(GId‘(1st ‘𝑅))}))) ∈ GrpOp)) |
6 | 5 | simplbi 499 | . 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
7 | eqid 2733 | . . 3 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
8 | 1, 2, 4, 3, 7 | dvrunz 36822 | . 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2nd ‘𝑅)) ≠ (GId‘(1st ‘𝑅))) |
9 | 1, 2, 4, 3 | divrngidl 36896 | . 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st ‘𝑅))}, ran (1st ‘𝑅)}) |
10 | 1, 2, 4, 3, 7 | smprngopr 36920 | . 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 1372 | 1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3946 {csn 4629 {cpr 4631 × cxp 5675 ran crn 5678 ↾ cres 5679 ‘cfv 6544 1st c1st 7973 2nd c2nd 7974 GrpOpcgr 29742 GIdcgi 29743 RingOpscrngo 36762 DivRingOpscdrng 36816 Idlcidl 36875 PrRingcprrng 36914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-1st 7975 df-2nd 7976 df-1o 8466 df-en 8940 df-grpo 29746 df-gid 29747 df-ginv 29748 df-ablo 29798 df-ass 36711 df-exid 36713 df-mgmOLD 36717 df-sgrOLD 36729 df-mndo 36735 df-rngo 36763 df-drngo 36817 df-idl 36878 df-pridl 36879 df-prrngo 36916 |
This theorem is referenced by: flddmn 36926 |
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