Theorem dvrunz 10211

Description: In a division ring the unit is different from the zero. (Contributed by FL, 14-Feb-2010.)

Hypotheses

Ref	Expression
dvrunz.1	|- G = (1st` R)
dvrunz.2	|- H = (2nd` R)
dvrunz.3	|- X = ran G
dvrunz.4	|- Z = (Id` G)
dvrunz.5	|- U = (Id` H)

Assertion

Ref	Expression
dvrunz	|- (R e. DivRing -> U =/= Z)

Proof of Theorem dvrunz

Step	Hyp	Ref	Expression
1		dvrunz.4	. . . 4 |- Z = (Id` G)
2		fvex 4500	. . . 4 |- (Id` G) e. _V
3	1, 2	eqeltri 1804	. . 3 |- Z e. _V
4	3	zrdivrng 10210	. 2 |- -. <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing
5		dvrunz.1	. . . . . . 7 |- G = (1st` R)
6		dvrunz.2	. . . . . . 7 |- H = (2nd` R)
7		dvrunz.3	. . . . . . 7 |- X = ran G
8	5, 6, 7, 1	drngi 9288	. . . . . 6 |- (R e. DivRing -> (R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
9	8	pm3.26d 346	. . . . 5 |- (R e. DivRing -> R e. Ring)
10		dvrunz.5	. . . . . 6 |- U = (Id` H)
11	5, 6, 1, 10, 7	uznzr 10208	. . . . 5 |- (R e. Ring -> (X ~~ 1o <-> U = Z))
12	9, 11	syl 12	. . . 4 |- (R e. DivRing -> (X ~~ 1o <-> U = Z))
13	5, 7, 1	on1el6 10206	. . . . . . 7 |- (R e. Ring -> (X ~~ 1o <-> R = <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>.))
14	9, 13	syl 12	. . . . . 6 |- (R e. DivRing -> (X ~~ 1o <-> R = <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>.))
15		eleq1 1794	. . . . . . 7 |- (R = <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. -> (R e. DivRing <-> <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing))
16	15	biimpd 169	. . . . . 6 |- (R = <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. -> (R e. DivRing -> <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing))
17	14, 16	syl6bi 230	. . . . 5 |- (R e. DivRing -> (X ~~ 1o -> (R e. DivRing -> <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing)))
18	17	pm2.43a 80	. . . 4 |- (R e. DivRing -> (X ~~ 1o -> <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing))
19	12, 18	sylbird 221	. . 3 |- (R e. DivRing -> (U = Z -> <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing))
20	19	necon3bd 1874	. 2 |- (R e. DivRing -> (-. <.{<.<.Z, Z>., Z>.}, {<.<.Z, Z>., Z>.}>. e. DivRing -> U =/= Z))
21	4, 20	mpi 55	1 |- (R e. DivRing -> U =/= Z)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 162   = wceq 1136   e. wcel 1138   =/= wne 1854  _Vcvv 2125   \ cdif 2423  {csn 2868  <.cop 2870   class class class wbr 3158   X. cxp 3795  ran crn 3798   |` cres 3799  ` cfv 3809  1stc1st 4829  2ndc2nd 4830  1oc1o 4983   ~~ cen 5234  Grpcgr 9106  Idcgi 9107  Ringcring 9258  DivRingcdrng 9286

This theorem is referenced by:  isdivrng2 15793  divrngpr 15883  isfldidl 15898

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-rep 3243  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-ral 1943  df-rex 1944  df-reu 1945  df-rab 1946  df-v 2127  df-sbc 2287  df-csb 2374  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-id 3401  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761  df-xp 3811  df-rel 3812  df-cnv 3813  df-co 3814  df-dm 3815  df-rn 3816  df-res 3817  df-ima 3818  df-fun 3819  df-fn 3820  df-f 3821  df-f1 3822  df-fo 3823  df-f1o 3824  df-fv 3825  df-opr 4697  df-1st 4831  df-2nd 4832  df-1o 4988  df-er 5129  df-en 5238  df-dom 5239  df-sdom 5240  df-fin 5241  df-grp 9111  df-gid 9112  df-abl 9203  df-ring 9259  df-drng 9287  df-ass 10152  df-exid 10154  df-mgm 10158  df-sgr 10170  df-mnd 10177

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