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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 2622 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2622 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2622 | . . . 4 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
4 | eqid 2622 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
5 | 1, 2, 3, 4 | isdrngo1 33755 | . . 3 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd ‘𝑅) ↾ ((ran (1st ‘𝑅) ∖ {(GId‘(1st ‘𝑅))}) × (ran (1st ‘𝑅) ∖ {(GId‘(1st ‘𝑅))}))) ∈ GrpOp)) |
6 | 5 | simplbi 476 | . 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
7 | eqid 2622 | . . 3 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
8 | 1, 2, 4, 3, 7 | dvrunz 33753 | . 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2nd ‘𝑅)) ≠ (GId‘(1st ‘𝑅))) |
9 | 1, 2, 4, 3 | divrngidl 33827 | . 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st ‘𝑅))}, ran (1st ‘𝑅)}) |
10 | 1, 2, 4, 3, 7 | smprngopr 33851 | . 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 1326 | 1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 {cpr 4179 × cxp 5112 ran crn 5115 ↾ cres 5116 ‘cfv 5888 1st c1st 7166 2nd c2nd 7167 GrpOpcgr 27343 GIdcgi 27344 RingOpscrngo 33693 DivRingOpscdrng 33747 Idlcidl 33806 PrRingcprrng 33845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-drngo 33748 df-idl 33809 df-pridl 33810 df-prrngo 33847 |
This theorem is referenced by: flddmn 33857 |
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