Theorem isdrngo3 32210

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 32209	. 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 3557	. . . . . 6 |- ( x e. ( X \ { Z } ) -> x e. X )
8		difss 3562	. . . . . . . 8 |- ( X \ { Z } ) C_ X
9		ssrexv 3496	. . . . . . . 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 2688	. . . . . . . . . . . . . . . 16 |- ( ( y H x ) = U -> ( ( y H x ) =/= Z <-> U =/= Z ) )
12	11	biimparc 490	. . . . . . . . . . . . . . 15 |- ( ( U =/= Z /\ ( y H x ) = U ) -> ( y H x ) =/= Z )
13	3, 4, 1, 2	rngolz 26141	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RingOps /\ x e. X ) -> ( Z H x ) = Z )
14		oveq1 6302	. . . . . . . . . . . . . . . . . . 19 |- ( y = Z -> ( y H x ) = ( Z H x ) )
15	14	eqeq1d 2455	. . . . . . . . . . . . . . . . . 18 |- ( y = Z -> ( ( y H x ) = Z <-> ( Z H x ) = Z ) )
16	13, 15	syl5ibrcom 226	. . . . . . . . . . . . . . . . 17 |- ( ( R e. RingOps /\ x e. X ) -> ( y = Z -> ( y H x ) = Z ) )
17	16	necon3d 2647	. . . . . . . . . . . . . . . 16 |- ( ( R e. RingOps /\ x e. X ) -> ( ( y H x ) =/= Z -> y =/= Z ) )
18	17	imp 431	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ x e. X ) /\ ( y H x ) =/= Z ) -> y =/= Z )
19	12, 18	sylan2 477	. . . . . . . . . . . . . 14 |- ( ( ( R e. RingOps /\ x e. X ) /\ ( U =/= Z /\ ( y H x ) = U ) ) -> y =/= Z )
20	19	an4s 836	. . . . . . . . . . . . 13 |- ( ( ( R e. RingOps /\ U =/= Z ) /\ ( x e. X /\ ( y H x ) = U ) ) -> y =/= Z )
21	20	anassrs 654	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> y =/= Z )
22		pm3.2 449	. . . . . . . . . . . 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 4100	. . . . . . . . . . 11 |- ( y e. ( X \ { Z } ) <-> ( y e. X /\ y =/= Z ) )
25	23, 24	syl6ibr 231	. . . . . . . . . 10 |- ( ( ( ( R e. RingOps /\ U =/= Z ) /\ x e. X ) /\ ( y H x ) = U ) -> ( y e. X -> y e. ( X \ { Z } ) ) )
26	25	imdistanda 700	. . . . . . . . 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 452	. . . . . . . . 9 |- ( ( y e. X /\ ( y H x ) = U ) <-> ( ( y H x ) = U /\ y e. X ) )
28		ancom 452	. . . . . . . . 9 |- ( ( y e. ( X \ { Z } ) /\ ( y H x ) = U ) <-> ( ( y H x ) = U /\ y e. ( X \ { Z } ) ) )
29	26, 27, 28	3imtr4g 274	. . . . . . . 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 2860	. . . . . . 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 208	. . . . . 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 477	. . . . 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 2826	. . . 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 647	. . 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 643	. 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 253	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 188   /\ wa 371   = wceq 1446   e. wcel 1889   =/= wne 2624   A.wral 2739   E.wrex 2740   \ cdif 3403   C_ wss 3406   {csn 3970   ran crn 4838   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797  GIdcgi 25927   RingOpscrngo 26115   DivRingOpscdrng 26145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-om 6698  df-1st 6798  df-2nd 6799  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-grpo 25931  df-gid 25932  df-ginv 25933  df-ablo 26022  df-ass 26053  df-exid 26055  df-mgmOLD 26059  df-sgrOLD 26071  df-mndo 26078  df-rngo 26116  df-drngo 26146

This theorem is referenced by:  isfldidl  32313