Theorem isdrngd 18772

Description: Properties that determine a division ring. I (reciprocal) is normally dependent on x i.e. read it as I ( x )." (Contributed by NM, 2-Aug-2013.)

Hypotheses

Ref	Expression
isdrngd.b	|- ( ph -> B = ( Base ` R ) )
isdrngd.t	|- ( ph -> .x. = ( .r ` R ) )
isdrngd.z	|- ( ph -> .0. = ( 0g ` R ) )
isdrngd.u	|- ( ph -> .1. = ( 1r ` R ) )
isdrngd.r	|- ( ph -> R e. Ring )
isdrngd.n	|- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) =/= .0. )
isdrngd.o	|- ( ph -> .1. =/= .0. )
isdrngd.i	|- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I e. B )
isdrngd.j	|- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I =/= .0. )
isdrngd.k	|- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( I .x. x ) = .1. )

Assertion

Ref	Expression
isdrngd	|- ( ph -> R e. DivRing )

Distinct variable groups:   x, y, .0.   x, .1. , y   x, B, y   y, I   x, R, y   ph, x, y   x, .x. , y

Allowed substitution hint:   I( x)

Proof of Theorem isdrngd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isdrngd.r	. . 3 |- ( ph -> R e. Ring )
2		difss 3737	. . . . . 6 |- ( B \ { .0. } ) C_ B
3		isdrngd.b	. . . . . 6 |- ( ph -> B = ( Base ` R ) )
4	2, 3	syl5sseq 3653	. . . . 5 |- ( ph -> ( B \ { .0. } ) C_ ( Base ` R ) )
5		eqid 2622	. . . . . 6 |- ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) )
6		eqid 2622	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
7		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8	6, 7	mgpbas 18495	. . . . . 6 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
9	5, 8	ressbas2 15931	. . . . 5 |- ( ( B \ { .0. } ) C_ ( Base ` R ) -> ( B \ { .0. } ) = ( Base ` ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) ) )
10	4, 9	syl 17	. . . 4 |- ( ph -> ( B \ { .0. } ) = ( Base ` ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) ) )
11		isdrngd.t	. . . . 5 |- ( ph -> .x. = ( .r ` R ) )
12		fvex 6201	. . . . . . 7 |- ( Base ` R ) e. _V
13	3, 12	syl6eqel 2709	. . . . . 6 |- ( ph -> B e. _V )
14		difexg 4808	. . . . . 6 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
15		eqid 2622	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
16	6, 15	mgpplusg 18493	. . . . . . 7 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
17	5, 16	ressplusg 15993	. . . . . 6 |- ( ( B \ { .0. } ) e. _V -> ( .r ` R ) = ( +g ` ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) ) )
18	13, 14, 17	3syl 18	. . . . 5 |- ( ph -> ( .r ` R ) = ( +g ` ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) ) )
19	11, 18	eqtrd 2656	. . . 4 |- ( ph -> .x. = ( +g ` ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) ) )
20		eldifsn 4317	. . . . 5 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
21		eldifsn 4317	. . . . . 6 |- ( y e. ( B \ { .0. } ) <-> ( y e. B /\ y =/= .0. ) )
22	7, 15	ringcl 18561	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) e. ( Base ` R ) )
23	1, 22	syl3an1 1359	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) e. ( Base ` R ) )
24	23	3expib 1268	. . . . . . . . . . 11 |- ( ph -> ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) e. ( Base ` R ) ) )
25	3	eleq2d 2687	. . . . . . . . . . . 12 |- ( ph -> ( x e. B <-> x e. ( Base ` R ) ) )
26	3	eleq2d 2687	. . . . . . . . . . . 12 |- ( ph -> ( y e. B <-> y e. ( Base ` R ) ) )
27	25, 26	anbi12d 747	. . . . . . . . . . 11 |- ( ph -> ( ( x e. B /\ y e. B ) <-> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) )
28	11	oveqd 6667	. . . . . . . . . . . 12 |- ( ph -> ( x .x. y ) = ( x ( .r ` R ) y ) )
29	28, 3	eleq12d 2695	. . . . . . . . . . 11 |- ( ph -> ( ( x .x. y ) e. B <-> ( x ( .r ` R ) y ) e. ( Base ` R ) ) )
30	24, 27, 29	3imtr4d 283	. . . . . . . . . 10 |- ( ph -> ( ( x e. B /\ y e. B ) -> ( x .x. y ) e. B ) )
31	30	3impib 1262	. . . . . . . . 9 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B )
32	31	3adant2r 1321	. . . . . . . 8 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ y e. B ) -> ( x .x. y ) e. B )
33	32	3adant3r 1323	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) e. B )
34		isdrngd.n	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) =/= .0. )
35		eldifsn 4317	. . . . . . 7 |- ( ( x .x. y ) e. ( B \ { .0. } ) <-> ( ( x .x. y ) e. B /\ ( x .x. y ) =/= .0. ) )
36	33, 34, 35	sylanbrc 698	. . . . . 6 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) e. ( B \ { .0. } ) )
37	21, 36	syl3an3b 1364	. . . . 5 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ y e. ( B \ { .0. } ) ) -> ( x .x. y ) e. ( B \ { .0. } ) )
38	20, 37	syl3an2b 1363	. . . 4 |- ( ( ph /\ x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) -> ( x .x. y ) e. ( B \ { .0. } ) )
39		eldifi 3732	. . . . . 6 |- ( x e. ( B \ { .0. } ) -> x e. B )
40		eldifi 3732	. . . . . 6 |- ( y e. ( B \ { .0. } ) -> y e. B )
41		eldifi 3732	. . . . . 6 |- ( z e. ( B \ { .0. } ) -> z e. B )
42	39, 40, 41	3anim123i 1247	. . . . 5 |- ( ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) /\ z e. ( B \ { .0. } ) ) -> ( x e. B /\ y e. B /\ z e. B ) )
43	7, 15	ringass 18564	. . . . . . . . 9 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) )
44	43	ex 450	. . . . . . . 8 |- ( R e. Ring -> ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) ) )
45	1, 44	syl 17	. . . . . . 7 |- ( ph -> ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) ) )
46	3	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( z e. B <-> z e. ( Base ` R ) ) )
47	25, 26, 46	3anbi123d 1399	. . . . . . 7 |- ( ph -> ( ( x e. B /\ y e. B /\ z e. B ) <-> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) )
48		eqidd 2623	. . . . . . . . 9 |- ( ph -> z = z )
49	11, 28, 48	oveq123d 6671	. . . . . . . 8 |- ( ph -> ( ( x .x. y ) .x. z ) = ( ( x ( .r ` R ) y ) ( .r ` R ) z ) )
50		eqidd 2623	. . . . . . . . 9 |- ( ph -> x = x )
51	11	oveqd 6667	. . . . . . . . 9 |- ( ph -> ( y .x. z ) = ( y ( .r ` R ) z ) )
52	11, 50, 51	oveq123d 6671	. . . . . . . 8 |- ( ph -> ( x .x. ( y .x. z ) ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) )
53	49, 52	eqeq12d 2637	. . . . . . 7 |- ( ph -> ( ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) <-> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) ) )
54	45, 47, 53	3imtr4d 283	. . . . . 6 |- ( ph -> ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) ) )
55	54	imp 445	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )
56	42, 55	sylan2 491	. . . 4 |- ( ( ph /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) /\ z e. ( B \ { .0. } ) ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )
57		eqid 2622	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
58	7, 57	ringidcl 18568	. . . . . . 7 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
59	1, 58	syl 17	. . . . . 6 |- ( ph -> ( 1r ` R ) e. ( Base ` R ) )
60		isdrngd.u	. . . . . 6 |- ( ph -> .1. = ( 1r ` R ) )
61	59, 60, 3	3eltr4d 2716	. . . . 5 |- ( ph -> .1. e. B )
62		isdrngd.o	. . . . 5 |- ( ph -> .1. =/= .0. )
63		eldifsn 4317	. . . . 5 |- ( .1. e. ( B \ { .0. } ) <-> ( .1. e. B /\ .1. =/= .0. ) )
64	61, 62, 63	sylanbrc 698	. . . 4 |- ( ph -> .1. e. ( B \ { .0. } ) )
65	7, 15, 57	ringlidm 18571	. . . . . . . . . 10 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x )
66	65	ex 450	. . . . . . . . 9 |- ( R e. Ring -> ( x e. ( Base ` R ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x ) )
67	1, 66	syl 17	. . . . . . . 8 |- ( ph -> ( x e. ( Base ` R ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x ) )
68	11, 60, 50	oveq123d 6671	. . . . . . . . 9 |- ( ph -> ( .1. .x. x ) = ( ( 1r ` R ) ( .r ` R ) x ) )
69	68	eqeq1d 2624	. . . . . . . 8 |- ( ph -> ( ( .1. .x. x ) = x <-> ( ( 1r ` R ) ( .r ` R ) x ) = x ) )
70	67, 25, 69	3imtr4d 283	. . . . . . 7 |- ( ph -> ( x e. B -> ( .1. .x. x ) = x ) )
71	70	imp 445	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x )
72	71	adantrr 753	. . . . 5 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( .1. .x. x ) = x )
73	20, 72	sylan2b 492	. . . 4 |- ( ( ph /\ x e. ( B \ { .0. } ) ) -> ( .1. .x. x ) = x )
74		isdrngd.i	. . . . . 6 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I e. B )
75		isdrngd.j	. . . . . 6 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I =/= .0. )
76		eldifsn 4317	. . . . . 6 |- ( I e. ( B \ { .0. } ) <-> ( I e. B /\ I =/= .0. ) )
77	74, 75, 76	sylanbrc 698	. . . . 5 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I e. ( B \ { .0. } ) )
78	20, 77	sylan2b 492	. . . 4 |- ( ( ph /\ x e. ( B \ { .0. } ) ) -> I e. ( B \ { .0. } ) )
79		isdrngd.k	. . . . 5 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( I .x. x ) = .1. )
80	20, 79	sylan2b 492	. . . 4 |- ( ( ph /\ x e. ( B \ { .0. } ) ) -> ( I .x. x ) = .1. )
81	10, 19, 38, 56, 64, 73, 78, 80	isgrpd 17444	. . 3 |- ( ph -> ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) e. Grp )
82		isdrngd.z	. . . . . . . 8 |- ( ph -> .0. = ( 0g ` R ) )
83	82	sneqd 4189	. . . . . . 7 |- ( ph -> { .0. } = { ( 0g ` R ) } )
84	3, 83	difeq12d 3729	. . . . . 6 |- ( ph -> ( B \ { .0. } ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) )
85	84	oveq2d 6666	. . . . 5 |- ( ph -> ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) = ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) )
86	85	eleq1d 2686	. . . 4 |- ( ph -> ( ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) e. Grp <-> ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) )
87	86	anbi2d 740	. . 3 |- ( ph -> ( ( R e. Ring /\ ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) e. Grp ) <-> ( R e. Ring /\ ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) ) )
88	1, 81, 87	mpbi2and 956	. 2 |- ( ph -> ( R e. Ring /\ ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) )
89		eqid 2622	. . 3 |- ( 0g ` R ) = ( 0g ` R )
90		eqid 2622	. . 3 |- ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) = ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) )
91	7, 89, 90	isdrng2 18757	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ ( (mulGrp ` R ) ↾s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) )
92	88, 91	sylibr 224	1 |- ( ph -> R e. DivRing )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   DivRingcdr 18747

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-pred 5680  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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749

This theorem is referenced by:  isdrngrd  18773  cndrng  19775  erngdvlem4  36279