divrngpr - Mathbox for Jeff Madsen

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Theorem divrngpr 38704

Description: Obsolete theorem, use drngprmrng 49105 instead. A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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

**Proof of Theorem divrngpr**
Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2763	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		eqid 2763	. . . 4 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
4		eqid 2763	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
5	1, 2, 3, 4	isdrngo1 38607	. . 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 501	. 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7		eqid 2763	. . 3 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 2, 4, 3, 7	dvrunz 38605	. 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2^nd ‘𝑅)) ≠ (GId‘(1^st ‘𝑅)))
9	1, 2, 4, 3	divrngidl 38679	. 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1^st ‘𝑅))}, ran (1^st ‘𝑅)})
10	1, 2, 4, 3, 7	smprngopr 38703	. 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 1398	1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3902 {csn 4589 {cpr 4591 × cxp 5659 ran crn 5662 ↾ cres 5663 ‘cfv 6536 1^st c1st 7980 2^nd c2nd 7981 GrpOpcgr 30841 GIdcgi 30842 RingOpscrngo 38545 DivRingOpscdrng 38599 Idlcidl 38658 PrRingcprrng 38697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-1st 7982 df-2nd 7983 df-1o 8449 df-en 8940 df-grpo 30845 df-gid 30846 df-ginv 30847 df-ablo 30897 df-ass 38494 df-exid 38496 df-mgmOLD 38500 df-sgrOLD 38512 df-mndo 38518 df-rngo 38546 df-drngo 38600 df-idl 38661 df-pridl 38662 df-prrngo 38699

This theorem is referenced by: flddmn 38709

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