Theorem isdrngo3 30567

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 30566	. 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 3622	. . . . . 6 |- ( x e. ( X \ { Z } ) -> x e. X )
8		difss 3627	. . . . . . . 8 |- ( X \ { Z } ) C_ X
9		ssrexv 3561	. . . . . . . 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 2738	. . . . . . . . . . . . . . . 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 25530	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RingOps /\ x e. X ) -> ( Z H x ) = Z )
14		oveq1 6303	. . . . . . . . . . . . . . . . . . 19 |- ( y = Z -> ( y H x ) = ( Z H x ) )
15	14	eqeq1d 2459	. . . . . . . . . . . . . . . . . 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 2681	. . . . . . . . . . . . . . . 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 826	. . . . . . . . . . . . 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 4157	. . . . . . . . . . 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 2928	. . . . . . 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 2893	. . . 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 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   E.wrex 2808   \ cdif 3468   C_ wss 3471   {csn 4032   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  GIdcgi 25316   RingOpscrngo 25504   DivRingOpscdrng 25534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-grpo 25320  df-gid 25321  df-ginv 25322  df-ablo 25411  df-ass 25442  df-exid 25444  df-mgmOLD 25448  df-sgrOLD 25460  df-mndo 25467  df-rngo 25505  df-drngo 25535

This theorem is referenced by:  isfldidl  30670