Theorem rngurd 29788

Description: Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)

Hypotheses

Ref	Expression
rngurd.b	|- ( ph -> B = ( Base ` R ) )
rngurd.p	|- ( ph -> .x. = ( .r ` R ) )
rngurd.z	|- ( ph -> .1. e. B )
rngurd.i	|- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x )
rngurd.j	|- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x )

Assertion

Ref	Expression
rngurd	|- ( ph -> .1. = ( 1r ` R ) )

Distinct variable groups:   x, B   x, R   x, .1.   x, .x.   ph, x

Proof of Theorem rngurd

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
3		eqid 2622	. . 3 |- ( 1r ` R ) = ( 1r ` R )
4	1, 2, 3	dfur2 18504	. 2 |- ( 1r ` R ) = ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) )
5		rngurd.z	. . . 4 |- ( ph -> .1. e. B )
6		rngurd.b	. . . 4 |- ( ph -> B = ( Base ` R ) )
7	5, 6	eleqtrd 2703	. . 3 |- ( ph -> .1. e. ( Base ` R ) )
8		rngurd.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x )
9		rngurd.j	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x )
10	8, 9	jca 554	. . . . 5 |- ( ( ph /\ x e. B ) -> ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) )
11	10	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. B ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) )
12		rngurd.p	. . . . . . . . 9 |- ( ph -> .x. = ( .r ` R ) )
13	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. B ) -> .x. = ( .r ` R ) )
14	13	oveqd 6667	. . . . . . 7 |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = ( .1. ( .r ` R ) x ) )
15	14	eqeq1d 2624	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( ( .1. .x. x ) = x <-> ( .1. ( .r ` R ) x ) = x ) )
16	13	oveqd 6667	. . . . . . 7 |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = ( x ( .r ` R ) .1. ) )
17	16	eqeq1d 2624	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( ( x .x. .1. ) = x <-> ( x ( .r ` R ) .1. ) = x ) )
18	15, 17	anbi12d 747	. . . . 5 |- ( ( ph /\ x e. B ) -> ( ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) <-> ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) )
19	6, 18	raleqbidva 3154	. . . 4 |- ( ph -> ( A. x e. B ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) <-> A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) )
20	11, 19	mpbid 222	. . 3 |- ( ph -> A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) )
21	6	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( e e. B <-> e e. ( Base ` R ) ) )
22	13	oveqd 6667	. . . . . . . . . . 11 |- ( ( ph /\ x e. B ) -> ( e .x. x ) = ( e ( .r ` R ) x ) )
23	22	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ph /\ x e. B ) -> ( ( e .x. x ) = x <-> ( e ( .r ` R ) x ) = x ) )
24	13	oveqd 6667	. . . . . . . . . . 11 |- ( ( ph /\ x e. B ) -> ( x .x. e ) = ( x ( .r ` R ) e ) )
25	24	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ph /\ x e. B ) -> ( ( x .x. e ) = x <-> ( x ( .r ` R ) e ) = x ) )
26	23, 25	anbi12d 747	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> ( ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) )
27	6, 26	raleqbidva 3154	. . . . . . . 8 |- ( ph -> ( A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) )
28	21, 27	anbi12d 747	. . . . . . 7 |- ( ph -> ( ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) <-> ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) )
29	8	ralrimiva 2966	. . . . . . . . . . . 12 |- ( ph -> A. x e. B ( .1. .x. x ) = x )
30	29	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> A. x e. B ( .1. .x. x ) = x )
31		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ e e. B ) -> e e. B )
32		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ e e. B ) /\ x = e ) -> x = e )
33	32	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ph /\ e e. B ) /\ x = e ) -> ( .1. .x. x ) = ( .1. .x. e ) )
34	33, 32	eqeq12d 2637	. . . . . . . . . . . 12 |- ( ( ( ph /\ e e. B ) /\ x = e ) -> ( ( .1. .x. x ) = x <-> ( .1. .x. e ) = e ) )
35	31, 34	rspcdv 3312	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> ( A. x e. B ( .1. .x. x ) = x -> ( .1. .x. e ) = e ) )
36	30, 35	mpd 15	. . . . . . . . . 10 |- ( ( ph /\ e e. B ) -> ( .1. .x. e ) = e )
37	36	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> ( .1. .x. e ) = e )
38	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> .1. e. B )
39		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) )
40		oveq2 6658	. . . . . . . . . . . . . 14 |- ( x = .1. -> ( e .x. x ) = ( e .x. .1. ) )
41		id 22	. . . . . . . . . . . . . 14 |- ( x = .1. -> x = .1. )
42	40, 41	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( x = .1. -> ( ( e .x. x ) = x <-> ( e .x. .1. ) = .1. ) )
43		oveq1 6657	. . . . . . . . . . . . . 14 |- ( x = .1. -> ( x .x. e ) = ( .1. .x. e ) )
44	43, 41	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( x = .1. -> ( ( x .x. e ) = x <-> ( .1. .x. e ) = .1. ) )
45	42, 44	anbi12d 747	. . . . . . . . . . . 12 |- ( x = .1. -> ( ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> ( ( e .x. .1. ) = .1. /\ ( .1. .x. e ) = .1. ) ) )
46	45	rspcva 3307	. . . . . . . . . . 11 |- ( ( .1. e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) -> ( ( e .x. .1. ) = .1. /\ ( .1. .x. e ) = .1. ) )
47	46	simprd 479	. . . . . . . . . 10 |- ( ( .1. e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) -> ( .1. .x. e ) = .1. )
48	38, 39, 47	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> ( .1. .x. e ) = .1. )
49	37, 48	eqtr3d 2658	. . . . . . . 8 |- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> e = .1. )
50	49	ex 450	. . . . . . 7 |- ( ph -> ( ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) -> e = .1. ) )
51	28, 50	sylbird 250	. . . . . 6 |- ( ph -> ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) -> e = .1. ) )
52	51	alrimiv 1855	. . . . 5 |- ( ph -> A. e ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) -> e = .1. ) )
53		eleq1 2689	. . . . . . 7 |- ( e = .1. -> ( e e. ( Base ` R ) <-> .1. e. ( Base ` R ) ) )
54		oveq1 6657	. . . . . . . . . 10 |- ( e = .1. -> ( e ( .r ` R ) x ) = ( .1. ( .r ` R ) x ) )
55	54	eqeq1d 2624	. . . . . . . . 9 |- ( e = .1. -> ( ( e ( .r ` R ) x ) = x <-> ( .1. ( .r ` R ) x ) = x ) )
56		oveq2 6658	. . . . . . . . . 10 |- ( e = .1. -> ( x ( .r ` R ) e ) = ( x ( .r ` R ) .1. ) )
57	56	eqeq1d 2624	. . . . . . . . 9 |- ( e = .1. -> ( ( x ( .r ` R ) e ) = x <-> ( x ( .r ` R ) .1. ) = x ) )
58	55, 57	anbi12d 747	. . . . . . . 8 |- ( e = .1. -> ( ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) <-> ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) )
59	58	ralbidv 2986	. . . . . . 7 |- ( e = .1. -> ( A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) <-> A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) )
60	53, 59	anbi12d 747	. . . . . 6 |- ( e = .1. -> ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) <-> ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) ) )
61	60	eqeu 3377	. . . . 5 |- ( ( .1. e. ( Base ` R ) /\ ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) /\ A. e ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) -> e = .1. ) ) -> E! e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) )
62	7, 7, 20, 52, 61	syl121anc 1331	. . . 4 |- ( ph -> E! e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) )
63	60	iota2 5877	. . . 4 |- ( ( .1. e. B /\ E! e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) -> ( ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) <-> ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) = .1. ) )
64	5, 62, 63	syl2anc 693	. . 3 |- ( ph -> ( ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) <-> ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) = .1. ) )
65	7, 20, 64	mpbi2and 956	. 2 |- ( ph -> ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) = .1. )
66	4, 65	syl5req 2669	1 |- ( ph -> .1. = ( 1r ` R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E!weu 2470   A.wral 2912   iotacio 5849   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   1rcur 18501

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-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-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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgp 18490  df-ur 18502

This theorem is referenced by:  ress1r  29789