Theorem drngnidl 19229

Description: A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)

Hypotheses

Ref	Expression
drngnidl.b	|- B = ( Base ` R )
drngnidl.z	|- .0. = ( 0g ` R )
drngnidl.u	|- U = (LIdeal ` R )

Assertion

Ref	Expression
drngnidl	|- ( R e. DivRing -> U = { { .0. } , B } )

Proof of Theorem drngnidl

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( ( R e. DivRing /\ a e. U ) /\ a = { .0. } ) -> a = { .0. } )
2	1	orcd 407	. . . . . 6 |- ( ( ( R e. DivRing /\ a e. U ) /\ a = { .0. } ) -> ( a = { .0. } \/ a = B ) )
3		drngring 18754	. . . . . . . . . . 11 |- ( R e. DivRing -> R e. Ring )
4	3	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> R e. Ring )
5		simplr 792	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> a e. U )
6		simpr 477	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> a =/= { .0. } )
7		drngnidl.u	. . . . . . . . . . 11 |- U = (LIdeal ` R )
8		drngnidl.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
9	7, 8	lidlnz 19228	. . . . . . . . . 10 |- ( ( R e. Ring /\ a e. U /\ a =/= { .0. } ) -> E. b e. a b =/= .0. )
10	4, 5, 6, 9	syl3anc 1326	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> E. b e. a b =/= .0. )
11		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> R e. DivRing )
12		drngnidl.b	. . . . . . . . . . . . . . . . 17 |- B = ( Base ` R )
13	12, 7	lidlss 19210	. . . . . . . . . . . . . . . 16 |- ( a e. U -> a C_ B )
14	13	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( R e. DivRing /\ a e. U ) -> a C_ B )
15	14	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ( R e. DivRing /\ a e. U ) /\ b e. a ) -> b e. B )
16	15	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> b e. B )
17		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> b =/= .0. )
18		eqid 2622	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( .r ` R )
19		eqid 2622	. . . . . . . . . . . . . 14 |- ( 1r ` R ) = ( 1r ` R )
20		eqid 2622	. . . . . . . . . . . . . 14 |- ( invr ` R ) = ( invr ` R )
21	12, 8, 18, 19, 20	drnginvrl 18766	. . . . . . . . . . . . 13 |- ( ( R e. DivRing /\ b e. B /\ b =/= .0. ) -> ( ( ( invr ` R ) ` b ) ( .r ` R ) b ) = ( 1r ` R ) )
22	11, 16, 17, 21	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> ( ( ( invr ` R ) ` b ) ( .r ` R ) b ) = ( 1r ` R ) )
23	3	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> R e. Ring )
24		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> a e. U )
25	12, 8, 20	drnginvrcl 18764	. . . . . . . . . . . . . 14 |- ( ( R e. DivRing /\ b e. B /\ b =/= .0. ) -> ( ( invr ` R ) ` b ) e. B )
26	11, 16, 17, 25	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> ( ( invr ` R ) ` b ) e. B )
27		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> b e. a )
28	7, 12, 18	lidlmcl 19217	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ a e. U ) /\ ( ( ( invr ` R ) ` b ) e. B /\ b e. a ) ) -> ( ( ( invr ` R ) ` b ) ( .r ` R ) b ) e. a )
29	23, 24, 26, 27, 28	syl22anc 1327	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> ( ( ( invr ` R ) ` b ) ( .r ` R ) b ) e. a )
30	22, 29	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ a e. U ) /\ ( b e. a /\ b =/= .0. ) ) -> ( 1r ` R ) e. a )
31	30	rexlimdvaa 3032	. . . . . . . . . 10 |- ( ( R e. DivRing /\ a e. U ) -> ( E. b e. a b =/= .0. -> ( 1r ` R ) e. a ) )
32	31	imp 445	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ a e. U ) /\ E. b e. a b =/= .0. ) -> ( 1r ` R ) e. a )
33	10, 32	syldan 487	. . . . . . . 8 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> ( 1r ` R ) e. a )
34	7, 12, 19	lidl1el 19218	. . . . . . . . . 10 |- ( ( R e. Ring /\ a e. U ) -> ( ( 1r ` R ) e. a <-> a = B ) )
35	3, 34	sylan 488	. . . . . . . . 9 |- ( ( R e. DivRing /\ a e. U ) -> ( ( 1r ` R ) e. a <-> a = B ) )
36	35	adantr 481	. . . . . . . 8 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> ( ( 1r ` R ) e. a <-> a = B ) )
37	33, 36	mpbid 222	. . . . . . 7 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> a = B )
38	37	olcd 408	. . . . . 6 |- ( ( ( R e. DivRing /\ a e. U ) /\ a =/= { .0. } ) -> ( a = { .0. } \/ a = B ) )
39	2, 38	pm2.61dane 2881	. . . . 5 |- ( ( R e. DivRing /\ a e. U ) -> ( a = { .0. } \/ a = B ) )
40		vex 3203	. . . . . 6 |- a e. _V
41	40	elpr 4198	. . . . 5 |- ( a e. { { .0. } , B } <-> ( a = { .0. } \/ a = B ) )
42	39, 41	sylibr 224	. . . 4 |- ( ( R e. DivRing /\ a e. U ) -> a e. { { .0. } , B } )
43	42	ex 450	. . 3 |- ( R e. DivRing -> ( a e. U -> a e. { { .0. } , B } ) )
44	43	ssrdv 3609	. 2 |- ( R e. DivRing -> U C_ { { .0. } , B } )
45	7, 8	lidl0 19219	. . . 4 |- ( R e. Ring -> { .0. } e. U )
46	7, 12	lidl1 19220	. . . 4 |- ( R e. Ring -> B e. U )
47		snex 4908	. . . . . 6 |- { .0. } e. _V
48		fvex 6201	. . . . . . 7 |- ( Base ` R ) e. _V
49	12, 48	eqeltri 2697	. . . . . 6 |- B e. _V
50	47, 49	prss 4351	. . . . 5 |- ( ( { .0. } e. U /\ B e. U ) <-> { { .0. } , B } C_ U )
51	50	bicomi 214	. . . 4 |- ( { { .0. } , B } C_ U <-> ( { .0. } e. U /\ B e. U ) )
52	45, 46, 51	sylanbrc 698	. . 3 |- ( R e. Ring -> { { .0. } , B } C_ U )
53	3, 52	syl 17	. 2 |- ( R e. DivRing -> { { .0. } , B } C_ U )
54	44, 53	eqssd 3620	1 |- ( R e. DivRing -> U = { { .0. } , B } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100   1rcur 18501   Ringcrg 18547   invrcinvr 18671   DivRingcdr 18747  LIdealclidl 19170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-lidl 19174

This theorem is referenced by:  drnglpir  19253