divrngpr - Mathbox for Jeff Madsen

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Theorem divrngpr 38047

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 2729	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2729	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		eqid 2729	. . . 4 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
4		eqid 2729	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
5	1, 2, 3, 4	isdrngo1 37950	. . 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 2729	. . 3 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 2, 4, 3, 7	dvrunz 37948	. 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2^nd ‘𝑅)) ≠ (GId‘(1^st ‘𝑅)))
9	1, 2, 4, 3	divrngidl 38022	. 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1^st ‘𝑅))}, ran (1^st ‘𝑅)})
10	1, 2, 4, 3, 7	smprngopr 38046	. 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 2925 ∖ cdif 3911 {csn 4589 {cpr 4591 × cxp 5636 ran crn 5639 ↾ cres 5640 ‘cfv 6511 1^st c1st 7966 2^nd c2nd 7967 GrpOpcgr 30418 GIdcgi 30419 RingOpscrngo 37888 DivRingOpscdrng 37942 Idlcidl 38001 PrRingcprrng 38040

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-1st 7968 df-2nd 7969 df-1o 8434 df-en 8919 df-grpo 30422 df-gid 30423 df-ginv 30424 df-ablo 30474 df-ass 37837 df-exid 37839 df-mgmOLD 37843 df-sgrOLD 37855 df-mndo 37861 df-rngo 37889 df-drngo 37943 df-idl 38004 df-pridl 38005 df-prrngo 38042

This theorem is referenced by: flddmn 38052

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