divrngpr - Mathbox for Jeff Madsen

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Theorem divrngpr 38040

Description: A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

**Assertion**
Ref	Expression
divrngpr	⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

**Proof of Theorem divrngpr**
Step	Hyp	Ref	Expression
1		eqid 2735	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2735	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		eqid 2735	. . . 4 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
4		eqid 2735	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
5	1, 2, 3, 4	isdrngo1 37943	. . 3 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2^nd ‘𝑅) ↾ ((ran (1^st ‘𝑅) ∖ {(GId‘(1^st ‘𝑅))}) × (ran (1^st ‘𝑅) ∖ {(GId‘(1^st ‘𝑅))}))) ∈ GrpOp))
6	5	simplbi 497	. 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7		eqid 2735	. . 3 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 2, 4, 3, 7	dvrunz 37941	. 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2^nd ‘𝑅)) ≠ (GId‘(1^st ‘𝑅)))
9	1, 2, 4, 3	divrngidl 38015	. 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1^st ‘𝑅))}, ran (1^st ‘𝑅)})
10	1, 2, 4, 3, 7	smprngopr 38039	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2^nd ‘𝑅)) ≠ (GId‘(1^st ‘𝑅)) ∧ (Idl‘𝑅) = {{(GId‘(1^st ‘𝑅))}, ran (1^st ‘𝑅)}) → 𝑅 ∈ PrRing)
11	6, 8, 9, 10	syl3anc 1370	1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 × cxp 5687 ran crn 5690 ↾ cres 5691 ‘cfv 6563 1^st c1st 8011 2^nd c2nd 8012 GrpOpcgr 30518 GIdcgi 30519 RingOpscrngo 37881 DivRingOpscdrng 37935 Idlcidl 37994 PrRingcprrng 38033

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-1st 8013 df-2nd 8014 df-1o 8505 df-en 8985 df-grpo 30522 df-gid 30523 df-ginv 30524 df-ablo 30574 df-ass 37830 df-exid 37832 df-mgmOLD 37836 df-sgrOLD 37848 df-mndo 37854 df-rngo 37882 df-drngo 37936 df-idl 37997 df-pridl 37998 df-prrngo 38035

This theorem is referenced by: flddmn 38045

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