divrngpr - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > divrngpr		Structured version Visualization version GIF version

Theorem divrngpr 38552

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 2762	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2762	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		eqid 2762	. . . 4 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
4		eqid 2762	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
5	1, 2, 3, 4	isdrngo1 38455	. . 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 500	. 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7		eqid 2762	. . 3 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 2, 4, 3, 7	dvrunz 38453	. 2 ⊢ (𝑅 ∈ DivRingOps → (GId‘(2^nd ‘𝑅)) ≠ (GId‘(1^st ‘𝑅)))
9	1, 2, 4, 3	divrngidl 38527	. 2 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1^st ‘𝑅))}, ran (1^st ‘𝑅)})
10	1, 2, 4, 3, 7	smprngopr 38551	. 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 1390	1 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∖ cdif 3901 {csn 4582 {cpr 4584 × cxp 5645 ran crn 5648 ↾ cres 5649 ‘cfv 6521 1^st c1st 7968 2^nd c2nd 7969 GrpOpcgr 30692 GIdcgi 30693 RingOpscrngo 38393 DivRingOpscdrng 38447 Idlcidl 38506 PrRingcprrng 38545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-1st 7970 df-2nd 7971 df-1o 8437 df-en 8928 df-grpo 30696 df-gid 30697 df-ginv 30698 df-ablo 30748 df-ass 38342 df-exid 38344 df-mgmOLD 38348 df-sgrOLD 38360 df-mndo 38366 df-rngo 38394 df-drngo 38448 df-idl 38509 df-pridl 38510 df-prrngo 38547

This theorem is referenced by: flddmn 38557

W3C validator