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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 2737 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2737 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2737 | . . . 4 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
4 | eqid 2737 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
5 | 1, 2, 3, 4 | isdrngo1 36418 | . . 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 2737 | . . 3 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
8 | 1, 2, 4, 3, 7 | dvrunz 36416 | . 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2nd ‘𝑅)) ≠ (GId‘(1st ‘𝑅))) |
9 | 1, 2, 4, 3 | divrngidl 36490 | . 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st ‘𝑅))}, ran (1st ‘𝑅)}) |
10 | 1, 2, 4, 3, 7 | smprngopr 36514 | . 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 2944 ∖ cdif 3908 {csn 4587 {cpr 4589 × cxp 5632 ran crn 5635 ↾ cres 5636 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 GrpOpcgr 29434 GIdcgi 29435 RingOpscrngo 36356 DivRingOpscdrng 36410 Idlcidl 36469 PrRingcprrng 36508 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-1st 7922 df-2nd 7923 df-1o 8413 df-en 8885 df-grpo 29438 df-gid 29439 df-ginv 29440 df-ablo 29490 df-ass 36305 df-exid 36307 df-mgmOLD 36311 df-sgrOLD 36323 df-mndo 36329 df-rngo 36357 df-drngo 36411 df-idl 36472 df-pridl 36473 df-prrngo 36510 |
This theorem is referenced by: flddmn 36520 |
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