Theorem isdrngo3 33758

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 33757	. 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 3732	. . . . . 6 |- ( x e. ( X \ { Z } ) -> x e. X )
8		difss 3737	. . . . . . . 8 |- ( X \ { Z } ) C_ X
9		ssrexv 3667	. . . . . . . 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 2856	. . . . . . . . . . . . . . . 16 |- ( ( y H x ) = U -> ( ( y H x ) =/= Z <-> U =/= Z ) )
12	11	biimparc 504	. . . . . . . . . . . . . . 15 |- ( ( U =/= Z /\ ( y H x ) = U ) -> ( y H x ) =/= Z )
13	3, 4, 1, 2	rngolz 33721	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RingOps /\ x e. X ) -> ( Z H x ) = Z )
14		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( y = Z -> ( y H x ) = ( Z H x ) )
15	14	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( y = Z -> ( ( y H x ) = Z <-> ( Z H x ) = Z ) )
16	13, 15	syl5ibrcom 237	. . . . . . . . . . . . . . . . 17 |- ( ( R e. RingOps /\ x e. X ) -> ( y = Z -> ( y H x ) = Z ) )
17	16	necon3d 2815	. . . . . . . . . . . . . . . 16 |- ( ( R e. RingOps /\ x e. X ) -> ( ( y H x ) =/= Z -> y =/= Z ) )
18	17	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ x e. X ) /\ ( y H x ) =/= Z ) -> y =/= Z )
19	12, 18	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ( R e. RingOps /\ x e. X ) /\ ( U =/= Z /\ ( y H x ) = U ) ) -> y =/= Z )
20	19	an4s 869	. . . . . . . . . . . . 13 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ ( x e. X /\ ( y H x ) = U ) ) -> y =/= Z )
21	20	anassrs 680	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> y =/= Z )
22		pm3.2 463	. . . . . . . . . . . 12 |- ( y e. X -> ( y =/= Z -> ( y e. X /\ y =/= Z ) ) )
23	21, 22	syl5com 31	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> ( y e. X -> ( y e. X /\ y =/= Z ) ) )
24		eldifsn 4317	. . . . . . . . . . 11 |- ( y e. ( X \ { Z } ) <-> ( y e. X /\ y =/= Z ) )
25	23, 24	syl6ibr 242	. . . . . . . . . 10 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> ( y e. X -> y e. ( X \ { Z } ) ) )
26	25	imdistanda 729	. . . . . . . . 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 466	. . . . . . . . 9 |- ( ( y e. X /\ ( y H x ) = U ) <-> ( ( y H x ) = U /\ y e. X ) )
28		ancom 466	. . . . . . . . 9 |- ( ( y e. ( X \ { Z } ) /\ ( y H x ) = U ) <-> ( ( y H x ) = U /\ y e. ( X \ { Z } ) ) )
29	26, 27, 28	3imtr4g 285	. . . . . . . 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 3014	. . . . . . 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 216	. . . . . 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 491	. . . . 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 2985	. . . 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 673	. . 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 669	. 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 264	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693   DivRingOpscdrng 33747

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

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-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-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-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-drngo 33748

This theorem is referenced by:  isfldidl  33867