divrngpr - Mathbox for Jeff Madsen

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Theorem divrngpr 38042

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 2730	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2730	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		eqid 2730	. . . 4 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
4		eqid 2730	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
5	1, 2, 3, 4	isdrngo1 37945	. . 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 2730	. . 3 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 2, 4, 3, 7	dvrunz 37943	. 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2^nd ‘𝑅)) ≠ (GId‘(1^st ‘𝑅)))
9	1, 2, 4, 3	divrngidl 38017	. 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1^st ‘𝑅))}, ran (1^st ‘𝑅)})
10	1, 2, 4, 3, 7	smprngopr 38041	. 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 1373	1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3913 {csn 4591 {cpr 4593 × cxp 5638 ran crn 5641 ↾ cres 5642 ‘cfv 6513 1^st c1st 7968 2^nd c2nd 7969 GrpOpcgr 30424 GIdcgi 30425 RingOpscrngo 37883 DivRingOpscdrng 37937 Idlcidl 37996 PrRingcprrng 38035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-1st 7970 df-2nd 7971 df-1o 8436 df-en 8921 df-grpo 30428 df-gid 30429 df-ginv 30430 df-ablo 30480 df-ass 37832 df-exid 37834 df-mgmOLD 37838 df-sgrOLD 37850 df-mndo 37856 df-rngo 37884 df-drngo 37938 df-idl 37999 df-pridl 38000 df-prrngo 38037

This theorem is referenced by: flddmn 38047

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