Theorem subrguss 18795

Description: A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrguss.1	|- S = ( R ↾s A )
subrguss.2	|- U = (Unit ` R )
subrguss.3	|- V = (Unit ` S )

Assertion

Ref	Expression
subrguss	|- ( A e. (SubRing ` R ) -> V C_ U )

Proof of Theorem subrguss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. V )
2		subrguss.3	. . . . . . . . 9 |- V = (Unit ` S )
3		eqid 2622	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
4		eqid 2622	. . . . . . . . 9 |- ( ||r ` S ) = ( ||r ` S )
5		eqid 2622	. . . . . . . . 9 |- (oppr ` S ) = (oppr ` S )
6		eqid 2622	. . . . . . . . 9 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
7	2, 3, 4, 5, 6	isunit 18657	. . . . . . . 8 |- ( x e. V <-> ( x ( ||r ` S ) ( 1r ` S ) /\ x ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) )
8	1, 7	sylib 208	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( ||r ` S ) ( 1r ` S ) /\ x ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) )
9	8	simpld 475	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` S ) )
10		subrguss.1	. . . . . . . 8 |- S = ( R ↾s A )
11		eqid 2622	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
12	10, 11	subrg1 18790	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
13	12	adantr 481	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( 1r ` R ) = ( 1r ` S ) )
14	9, 13	breqtrrd 4681	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` R ) )
15		eqid 2622	. . . . . . . 8 |- ( ||r ` R ) = ( ||r ` R )
16	10, 15, 4	subrgdvds 18794	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) C_ ( ||r ` R ) )
17	16	adantr 481	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` S ) C_ ( ||r ` R ) )
18	17	ssbrd 4696	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( ||r ` S ) ( 1r ` R ) -> x ( ||r ` R ) ( 1r ` R ) ) )
19	14, 18	mpd 15	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` R ) ( 1r ` R ) )
20	10	subrgbas 18789	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
21	20	adantr 481	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A = ( Base ` S ) )
22		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
23	22	subrgss 18781	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
24	23	adantr 481	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A C_ ( Base ` R ) )
25	21, 24	eqsstr3d 3640	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) C_ ( Base ` R ) )
26		eqid 2622	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
27	26, 2	unitcl 18659	. . . . . . . 8 |- ( x e. V -> x e. ( Base ` S ) )
28	27	adantl 482	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` S ) )
29	25, 28	sseldd 3604	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` R ) )
30	10	subrgring 18783	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> S e. Ring )
31		eqid 2622	. . . . . . . . 9 |- ( invr ` S ) = ( invr ` S )
32	2, 31, 26	ringinvcl 18676	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
33	30, 32	sylan 488	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
34	25, 33	sseldd 3604	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` R ) )
35		eqid 2622	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
36	35, 22	opprbas 18629	. . . . . . 7 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
37		eqid 2622	. . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
38		eqid 2622	. . . . . . 7 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
39	36, 37, 38	dvdsrmul 18648	. . . . . 6 |- ( ( x e. ( Base ` R ) /\ ( ( invr ` S ) ` x ) e. ( Base ` R ) ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) )
40	29, 34, 39	syl2anc 693	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) )
41		eqid 2622	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
42	22, 41, 35, 38	opprmul 18626	. . . . . 6 |- ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) )
43		eqid 2622	. . . . . . . . 9 |- ( .r ` S ) = ( .r ` S )
44	2, 31, 43, 3	unitrinv 18678	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) = ( 1r ` S ) )
45	30, 44	sylan 488	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) = ( 1r ` S ) )
46	10, 41	ressmulr 16006	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
47	46	adantr 481	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` R ) = ( .r ` S ) )
48	47	oveqd 6667	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) )
49	45, 48, 13	3eqtr4d 2666	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( 1r ` R ) )
50	42, 49	syl5eq 2668	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
51	40, 50	breqtrd 4679	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
52		subrguss.2	. . . . 5 |- U = (Unit ` R )
53	52, 11, 15, 35, 37	isunit 18657	. . . 4 |- ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
54	19, 51, 53	sylanbrc 698	. . 3 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. U )
55	54	ex 450	. 2 |- ( A e. (SubRing ` R ) -> ( x e. V -> x e. U ) )
56	55	ssrdv 3609	1 |- ( A e. (SubRing ` R ) -> V C_ U )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   .rcmulr 15942   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   ||_rcdsr 18638  Unitcui 18639   invrcinvr 18671  SubRingcsubrg 18776

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-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-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-subrg 18778

This theorem is referenced by:  subrginv  18796  subrgdv  18797  subrgunit  18798  subrgugrp  18799  issubdrg  18805  zringunit  19836