Theorem isdrngo3 29963

Description: A division ring is a ring in which 1 =/= 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

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

Assertion

Ref	Expression
isdrngo3	|- ( R e. DivRingOps <-> ( R e. RingOps /\ ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U ) ) )

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

Allowed substitution hints:   G( x, y)

Proof of Theorem isdrngo3

Step	Hyp	Ref	Expression
1		isdivrng1.1	. . 3 |- G = ( 1st ` R )
2		isdivrng1.2	. . 3 |- H = ( 2nd ` R )
3		isdivrng1.3	. . 3 |- Z = (GId ` G )
4		isdivrng1.4	. . 3 |- X = ran G
5		isdivrng2.5	. . 3 |- U = (GId ` H )
6	1, 2, 3, 4, 5	isdrngo2 29962	. 2 |- ( R e. DivRingOps <-> ( R e. RingOps /\ ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. ( X \ { Z } ) ( y H x ) = U ) ) )
7		eldifi 3626	. . . . . 6 |- ( x e. ( X \ { Z } ) -> x e. X )
8		difss 3631	. . . . . . . 8 |- ( X \ { Z } ) C_ X
9		ssrexv 3565	. . . . . . . 8 |- ( ( X \ { Z } ) C_ X -> ( E. y e. ( X \ { Z } ) ( y H x ) = U -> E. y e. X ( y H x ) = U ) )
10	8, 9	ax-mp 5	. . . . . . 7 |- ( E. y e. ( X \ { Z } ) ( y H x ) = U -> E. y e. X ( y H x ) = U )
11		neeq1 2748	. . . . . . . . . . . . . . . 16 |- ( ( y H x ) = U -> ( ( y H x ) =/= Z <-> U =/= Z ) )
12	11	biimparc 487	. . . . . . . . . . . . . . 15 |- ( ( U =/= Z /\ ( y H x ) = U ) -> ( y H x ) =/= Z )
13	3, 4, 1, 2	rngolz 25076	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RingOps /\ x e. X ) -> ( Z H x ) = Z )
14		oveq1 6289	. . . . . . . . . . . . . . . . . . 19 |- ( y = Z -> ( y H x ) = ( Z H x ) )
15	14	eqeq1d 2469	. . . . . . . . . . . . . . . . . 18 |- ( y = Z -> ( ( y H x ) = Z <-> ( Z H x ) = Z ) )
16	13, 15	syl5ibrcom 222	. . . . . . . . . . . . . . . . 17 |- ( ( R e. RingOps /\ x e. X ) -> ( y = Z -> ( y H x ) = Z ) )
17	16	necon3d 2691	. . . . . . . . . . . . . . . 16 |- ( ( R e. RingOps /\ x e. X ) -> ( ( y H x ) =/= Z -> y =/= Z ) )
18	17	imp 429	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ x e. X ) /\ ( y H x ) =/= Z ) -> y =/= Z )
19	12, 18	sylan2 474	. . . . . . . . . . . . . 14 |- ( ( ( R e. RingOps /\ x e. X ) /\ ( U =/= Z /\ ( y H x ) = U ) ) -> y =/= Z )
20	19	an4s 824	. . . . . . . . . . . . 13 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ ( x e. X /\ ( y H x ) = U ) ) -> y =/= Z )
21	20	anassrs 648	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> y =/= Z )
22		pm3.2 447	. . . . . . . . . . . 12 |- ( y e. X -> ( y =/= Z -> ( y e. X /\ y =/= Z ) ) )
23	21, 22	syl5com 30	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> ( y e. X -> ( y e. X /\ y =/= Z ) ) )
24		eldifsn 4152	. . . . . . . . . . 11 |- ( y e. ( X \ { Z } ) <-> ( y e. X /\ y =/= Z ) )
25	23, 24	syl6ibr 227	. . . . . . . . . 10 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> ( y e. X -> y e. ( X \ { Z } ) ) )
26	25	imdistanda 693	. . . . . . . . 9 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) -> ( ( ( y H x ) = U /\ y e. X ) -> ( ( y H x ) = U /\ y e. ( X \ { Z } ) ) ) )
27		ancom 450	. . . . . . . . 9 |- ( ( y e. X /\ ( y H x ) = U ) <-> ( ( y H x ) = U /\ y e. X ) )
28		ancom 450	. . . . . . . . 9 |- ( ( y e. ( X \ { Z } ) /\ ( y H x ) = U ) <-> ( ( y H x ) = U /\ y e. ( X \ { Z } ) ) )
29	26, 27, 28	3imtr4g 270	. . . . . . . 8 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) -> ( ( y e. X /\ ( y H x ) = U ) -> ( y e. ( X \ { Z } ) /\ ( y H x ) = U ) ) )
30	29	reximdv2 2934	. . . . . . 7 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) -> ( E. y e. X ( y H x ) = U -> E. y e. ( X \ { Z } ) ( y H x ) = U ) )
31	10, 30	impbid2 204	. . . . . 6 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) -> ( E. y e. ( X \ { Z } ) ( y H x ) = U <-> E. y e. X ( y H x ) = U ) )
32	7, 31	sylan2 474	. . . . 5 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. ( X \ { Z } ) ) -> ( E. y e. ( X \ { Z } ) ( y H x ) = U <-> E. y e. X ( y H x ) = U ) )
33	32	ralbidva 2900	. . . 4 |- ( ( R e. RingOps /\ U =/= Z ) -> ( A. x e. ( X \ { Z } ) E. y e. ( X \ { Z } ) ( y H x ) = U <-> A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U ) )
34	33	pm5.32da 641	. . 3 |- ( R e. RingOps -> ( ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. ( X \ { Z } ) ( y H x ) = U ) <-> ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U ) ) )
35	34	pm5.32i 637	. 2 |- ( ( R e. RingOps /\ ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. ( X \ { Z } ) ( y H x ) = U ) ) <-> ( R e. RingOps /\ ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U ) ) )
36	6, 35	bitri 249	1 |- ( R e. DivRingOps <-> ( R e. RingOps /\ ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   \ cdif 3473   C_ wss 3476   {csn 4027   ran crn 5000   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780  GIdcgi 24862   RingOpscrngo 25050   DivRingOpscdrng 25080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-grpo 24866  df-gid 24867  df-ginv 24868  df-ablo 24957  df-ass 24988  df-exid 24990  df-mgm 24994  df-sgr 25006  df-mndo 25013  df-rngo 25051  df-drngo 25081

This theorem is referenced by:  isfldidl  30066