Theorem isdrngo3 28933

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 28932	. 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 3589	. . . . . 6 |- ( x e. ( X \ { Z } ) -> x e. X )
8		difss 3594	. . . . . . . 8 |- ( X \ { Z } ) C_ X
9		ssrexv 3528	. . . . . . . 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 2733	. . . . . . . . . . . . . . . 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 24060	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RingOps /\ x e. X ) -> ( Z H x ) = Z )
14		oveq1 6210	. . . . . . . . . . . . . . . . . . 19 |- ( y = Z -> ( y H x ) = ( Z H x ) )
15	14	eqeq1d 2456	. . . . . . . . . . . . . . . . . 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 2676	. . . . . . . . . . . . . . . 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 822	. . . . . . . . . . . . 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 4111	. . . . . . . . . . 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 2931	. . . . . . 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 2844	. . . 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 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   \ cdif 3436   C_ wss 3439   {csn 3988   ran crn 4952   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689  GIdcgi 23846   RingOpscrngo 24034   DivRingOpscdrng 24064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-om 6590  df-1st 6690  df-2nd 6691  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-grpo 23850  df-gid 23851  df-ginv 23852  df-ablo 23941  df-ass 23972  df-exid 23974  df-mgm 23978  df-sgr 23990  df-mndo 23997  df-rngo 24035  df-drngo 24065

This theorem is referenced by:  isfldidl  29036