Theorem isdivrng3 16112

Description: A division ring is a ring in which 1 =/= 0 and every nonzero element is invertible.

Hypotheses

Ref	Expression
isdivrng1.1	|- G = (1st` R)
isdivrng1.2	|- H = (2nd` R)
isdivrng1.3	|- Z = (Id` G)
isdivrng1.4	|- X = ran G
isdivrng2.5	|- U = (Id` H)

Assertion

Ref	Expression
isdivrng3	|- (R e. DivRing <-> (R e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. X (yHx) = U)))

Distinct variable groups:   x,H,y   x,X,y   x,Z,y   x,R,y   x,U,y

Proof of Theorem isdivrng3

Step	Hyp	Ref	Expression
1		isdivrng1.1	. . 3 |- G = (1st` R)
2		isdivrng1.2	. . 3 |- H = (2nd` R)
3		isdivrng1.3	. . 3 |- Z = (Id` G)
4		isdivrng1.4	. . 3 |- X = ran G
5		isdivrng2.5	. . 3 |- U = (Id` H)
6	1, 2, 3, 4, 5	isdivrng2 16111	. 2 |- (R e. DivRing <-> (R e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = U)))
7		difss 2735	. . . . . . . 8 |- (X \ {Z}) C_ X
8		ssrexv 2673	. . . . . . . 8 |- ((X \ {Z}) C_ X -> (E.y e. (X \ {Z})(yHx) = U -> E.y e. X (yHx) = U))
9	7, 8	ax-mp 7	. . . . . . 7 |- (E.y e. (X \ {Z})(yHx) = U -> E.y e. X (yHx) = U)
10		pm3.2 305	. . . . . . . . . . . 12 |- (y e. X -> (y =/= Z -> (y e. X /\ y =/= Z)))
11		opreq1 4889	. . . . . . . . . . . . . . . . . . 19 |- (y = Z -> (yHx) = (ZHx))
12	11	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (y = Z -> ((yHx) = Z <-> (ZHx) = Z))
13	3, 4, 1, 2	ringlz 9487	. . . . . . . . . . . . . . . . . 18 |- ((R e. Ring /\ x e. X) -> (ZHx) = Z)
14	12, 13	syl5cbir 228	. . . . . . . . . . . . . . . . 17 |- ((R e. Ring /\ x e. X) -> (y = Z -> (yHx) = Z))
15	14	necon3d 2041	. . . . . . . . . . . . . . . 16 |- ((R e. Ring /\ x e. X) -> ((yHx) =/= Z -> y =/= Z))
16	15	imp 377	. . . . . . . . . . . . . . 15 |- (((R e. Ring /\ x e. X) /\ (yHx) =/= Z) -> y =/= Z)
17		neeq1 2024	. . . . . . . . . . . . . . . 16 |- ((yHx) = U -> ((yHx) =/= Z <-> U =/= Z))
18	17	biimparc 463	. . . . . . . . . . . . . . 15 |- ((U =/= Z /\ (yHx) = U) -> (yHx) =/= Z)
19	16, 18	sylan2 500	. . . . . . . . . . . . . 14 |- (((R e. Ring /\ x e. X) /\ (U =/= Z /\ (yHx) = U)) -> y =/= Z)
20	19	an4s 566	. . . . . . . . . . . . 13 |- (((R e. Ring /\ U =/= Z) /\ (x e. X /\ (yHx) = U)) -> y =/= Z)
21	20	anassrs 489	. . . . . . . . . . . 12 |- ((((R e. Ring /\ U =/= Z) /\ x e. X) /\ (yHx) = U) -> y =/= Z)
22	10, 21	syl5com 63	. . . . . . . . . . 11 |- ((((R e. Ring /\ U =/= Z) /\ x e. X) /\ (yHx) = U) -> (y e. X -> (y e. X /\ y =/= Z)))
23		eldifsn 3123	. . . . . . . . . . 11 |- (y e. (X \ {Z}) <-> (y e. X /\ y =/= Z))
24	22, 23	syl6ibr 230	. . . . . . . . . 10 |- ((((R e. Ring /\ U =/= Z) /\ x e. X) /\ (yHx) = U) -> (y e. X -> y e. (X \ {Z})))
25	24	imdistanda 15652	. . . . . . . . 9 |- (((R e. Ring /\ U =/= Z) /\ x e. X) -> (((yHx) = U /\ y e. X) -> ((yHx) = U /\ y e. (X \ {Z}))))
26		ancom 482	. . . . . . . . 9 |- ((y e. X /\ (yHx) = U) <-> ((yHx) = U /\ y e. X))
27		ancom 482	. . . . . . . . 9 |- ((y e. (X \ {Z}) /\ (yHx) = U) <-> ((yHx) = U /\ y e. (X \ {Z})))
28	25, 26, 27	3imtr4g 612	. . . . . . . 8 |- (((R e. Ring /\ U =/= Z) /\ x e. X) -> ((y e. X /\ (yHx) = U) -> (y e. (X \ {Z}) /\ (yHx) = U)))
29	28	reximdv2 2200	. . . . . . 7 |- (((R e. Ring /\ U =/= Z) /\ x e. X) -> (E.y e. X (yHx) = U -> E.y e. (X \ {Z})(yHx) = U))
30	9, 29	impbid2 576	. . . . . 6 |- (((R e. Ring /\ U =/= Z) /\ x e. X) -> (E.y e. (X \ {Z})(yHx) = U <-> E.y e. X (yHx) = U))
31		eldifi 2730	. . . . . 6 |- (x e. (X \ {Z}) -> x e. X)
32	30, 31	sylan2 500	. . . . 5 |- (((R e. Ring /\ U =/= Z) /\ x e. (X \ {Z})) -> (E.y e. (X \ {Z})(yHx) = U <-> E.y e. X (yHx) = U))
33	32	ralbidva 2119	. . . 4 |- ((R e. Ring /\ U =/= Z) -> (A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = U <-> A.x e. (X \ {Z})E.y e. X (yHx) = U))
34	33	pm5.32da 711	. . 3 |- (R e. Ring -> ((U =/= Z /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = U) <-> (U =/= Z /\ A.x e. (X \ {Z})E.y e. X (yHx) = U)))
35	34	pm5.32i 707	. 2 |- ((R e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = U)) <-> (R e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. X (yHx) = U)))
36	6, 35	bitri 190	1 |- (R e. DivRing <-> (R e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. X (yHx) = U)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   \ cdif 2590   C_ wss 2593  {csn 3044  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  DivRingcdrng 9491

This theorem is referenced by:  isfldidl 16216

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-1o 5177  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-ring 9464  df-drng 9492  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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