Theorem isdrngrd 18773

Description: Properties that determine a division ring. I (reciprocal) is normally dependent on x i.e. read it as I ( x )." This version of isdrngd 18772 requires a right reciprocal instead of left. (Contributed by NM, 10-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. )
isdrngrd.k	|- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( x .x. I ) = .1. )

Assertion

Ref	Expression
isdrngrd	|- ( 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 isdrngrd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isdrngd.b	. . . 4 |- ( ph -> B = ( Base ` R ) )
2		eqid 2622	. . . . 5 |- (oppr ` R ) = (oppr ` R )
3		eqid 2622	. . . . 5 |- ( Base ` R ) = ( Base ` R )
4	2, 3	opprbas 18629	. . . 4 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
5	1, 4	syl6eq 2672	. . 3 |- ( ph -> B = ( Base ` (oppr ` R ) ) )
6		eqidd 2623	. . 3 |- ( ph -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
7		isdrngd.z	. . . 4 |- ( ph -> .0. = ( 0g ` R ) )
8		eqid 2622	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
9	2, 8	oppr0 18633	. . . 4 |- ( 0g ` R ) = ( 0g ` (oppr ` R ) )
10	7, 9	syl6eq 2672	. . 3 |- ( ph -> .0. = ( 0g ` (oppr ` R ) ) )
11		isdrngd.u	. . . 4 |- ( ph -> .1. = ( 1r ` R ) )
12		eqid 2622	. . . . 5 |- ( 1r ` R ) = ( 1r ` R )
13	2, 12	oppr1 18634	. . . 4 |- ( 1r ` R ) = ( 1r ` (oppr ` R ) )
14	11, 13	syl6eq 2672	. . 3 |- ( ph -> .1. = ( 1r ` (oppr ` R ) ) )
15		isdrngd.r	. . . 4 |- ( ph -> R e. Ring )
16	2	opprring 18631	. . . 4 |- ( R e. Ring -> (oppr ` R ) e. Ring )
17	15, 16	syl 17	. . 3 |- ( ph -> (oppr ` R ) e. Ring )
18		eleq1 2689	. . . . . . 7 |- ( y = x -> ( y e. B <-> x e. B ) )
19		neeq1 2856	. . . . . . 7 |- ( y = x -> ( y =/= .0. <-> x =/= .0. ) )
20	18, 19	anbi12d 747	. . . . . 6 |- ( y = x -> ( ( y e. B /\ y =/= .0. ) <-> ( x e. B /\ x =/= .0. ) ) )
21	20	3anbi2d 1404	. . . . 5 |- ( y = x -> ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) <-> ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) ) )
22		oveq1 6657	. . . . . 6 |- ( y = x -> ( y ( .r ` (oppr ` R ) ) z ) = ( x ( .r ` (oppr ` R ) ) z ) )
23	22	neeq1d 2853	. . . . 5 |- ( y = x -> ( ( y ( .r ` (oppr ` R ) ) z ) =/= .0. <-> ( x ( .r ` (oppr ` R ) ) z ) =/= .0. ) )
24	21, 23	imbi12d 334	. . . 4 |- ( y = x -> ( ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( .r ` (oppr ` R ) ) z ) =/= .0. ) <-> ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( x ( .r ` (oppr ` R ) ) z ) =/= .0. ) ) )
25		eleq1 2689	. . . . . . . 8 |- ( x = z -> ( x e. B <-> z e. B ) )
26		neeq1 2856	. . . . . . . 8 |- ( x = z -> ( x =/= .0. <-> z =/= .0. ) )
27	25, 26	anbi12d 747	. . . . . . 7 |- ( x = z -> ( ( x e. B /\ x =/= .0. ) <-> ( z e. B /\ z =/= .0. ) ) )
28	27	3anbi3d 1405	. . . . . 6 |- ( x = z -> ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( x e. B /\ x =/= .0. ) ) <-> ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) ) )
29		oveq2 6658	. . . . . . 7 |- ( x = z -> ( y ( .r ` (oppr ` R ) ) x ) = ( y ( .r ` (oppr ` R ) ) z ) )
30	29	neeq1d 2853	. . . . . 6 |- ( x = z -> ( ( y ( .r ` (oppr ` R ) ) x ) =/= .0. <-> ( y ( .r ` (oppr ` R ) ) z ) =/= .0. ) )
31	28, 30	imbi12d 334	. . . . 5 |- ( x = z -> ( ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( x e. B /\ x =/= .0. ) ) -> ( y ( .r ` (oppr ` R ) ) x ) =/= .0. ) <-> ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( .r ` (oppr ` R ) ) z ) =/= .0. ) ) )
32		isdrngd.t	. . . . . . . . . 10 |- ( ph -> .x. = ( .r ` R ) )
33	32	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> .x. = ( .r ` R ) )
34	33	oveqd 6667	. . . . . . . 8 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) = ( x ( .r ` R ) y ) )
35		eqid 2622	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
36		eqid 2622	. . . . . . . . 9 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
37	3, 35, 2, 36	opprmul 18626	. . . . . . . 8 |- ( y ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) y )
38	34, 37	syl6eqr 2674	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) = ( y ( .r ` (oppr ` R ) ) x ) )
39		isdrngd.n	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x .x. y ) =/= .0. )
40	38, 39	eqnetrrd 2862	. . . . . 6 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( y ( .r ` (oppr ` R ) ) x ) =/= .0. )
41	40	3com23 1271	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( x e. B /\ x =/= .0. ) ) -> ( y ( .r ` (oppr ` R ) ) x ) =/= .0. )
42	31, 41	chvarv 2263	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( .r ` (oppr ` R ) ) z ) =/= .0. )
43	24, 42	chvarv 2263	. . 3 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( x ( .r ` (oppr ` R ) ) z ) =/= .0. )
44		isdrngd.o	. . 3 |- ( ph -> .1. =/= .0. )
45		isdrngd.i	. . 3 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I e. B )
46		isdrngd.j	. . 3 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> I =/= .0. )
47	3, 35, 2, 36	opprmul 18626	. . . 4 |- ( I ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) I )
48	32	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> .x. = ( .r ` R ) )
49	48	oveqd 6667	. . . . 5 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( x .x. I ) = ( x ( .r ` R ) I ) )
50		isdrngrd.k	. . . . 5 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( x .x. I ) = .1. )
51	49, 50	eqtr3d 2658	. . . 4 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( x ( .r ` R ) I ) = .1. )
52	47, 51	syl5eq 2668	. . 3 |- ( ( ph /\ ( x e. B /\ x =/= .0. ) ) -> ( I ( .r ` (oppr ` R ) ) x ) = .1. )
53	5, 6, 10, 14, 17, 43, 44, 45, 46, 52	isdrngd 18772	. 2 |- ( ph -> (oppr ` R ) e. DivRing )
54	2	opprdrng 18771	. 2 |- ( R e. DivRing <-> (oppr ` R ) e. DivRing )
55	53, 54	sylibr 224	1 |- ( ph -> R e. DivRing )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   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:  erngdvlem4-rN  36287