Theorem subrguss 14517

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		subrguss.3	. . . . . . . . 9 |- V = (Unit ` S )
2	1	a1i 9	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> V = (Unit ` S ) )
3		eqidd 2239	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
4		eqidd 2239	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
5		eqidd 2239	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
6		eqidd 2239	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
7		subrguss.1	. . . . . . . . . 10 |- S = ( R ↾s A )
8	7	subrgring 14505	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> S e. Ring )
9		ringsrg 14325	. . . . . . . . 9 |- ( S e. Ring -> S e. SRing )
10	8, 9	syl 14	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> S e. SRing )
11	2, 3, 4, 5, 6, 10	isunitd 14386	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( x e. V <-> ( x ( ||r ` S ) ( 1r ` S ) /\ x ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) ) )
12	11	simprbda 383	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` S ) )
13		eqid 2238	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
14	7, 13	subrg1 14512	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
15	14	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( 1r ` R ) = ( 1r ` S ) )
16	12, 15	breqtrrd 4153	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` R ) )
17		eqid 2238	. . . . . . . 8 |- ( ||r ` R ) = ( ||r ` R )
18		eqid 2238	. . . . . . . 8 |- ( ||r ` S ) = ( ||r ` S )
19	7, 17, 18	subrgdvds 14516	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) C_ ( ||r ` R ) )
20	19	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` S ) C_ ( ||r ` R ) )
21	20	ssbrd 4168	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( ||r ` S ) ( 1r ` R ) -> x ( ||r ` R ) ( 1r ` R ) ) )
22	16, 21	mpd 13	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` R ) ( 1r ` R ) )
23		subrgrcl 14507	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> R e. Ring )
24	23	adantr 276	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> R e. Ring )
25		eqid 2238	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
26		eqid 2238	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
27	25, 26	opprbasg 14353	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
28	24, 27	syl 14	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
29		eqidd 2239	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
30	25	opprring 14357	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
31		ringsrg 14325	. . . . . . 7 |- ( (oppr ` R ) e. Ring -> (oppr ` R ) e. SRing )
32	24, 30, 31	3syl 17	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> (oppr ` R ) e. SRing )
33		eqidd 2239	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
34	7	subrgbas 14511	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
35	34	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A = ( Base ` S ) )
36	26	subrgss 14503	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
37	36	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A C_ ( Base ` R ) )
38	35, 37	eqsstrrd 3285	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) C_ ( Base ` R ) )
39		eqidd 2239	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) = ( Base ` S ) )
40	1	a1i 9	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> V = (Unit ` S ) )
41	10	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> S e. SRing )
42		simpr 110	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. V )
43	39, 40, 41, 42	unitcld 14388	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` S ) )
44	38, 43	sseldd 3249	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` R ) )
45		eqid 2238	. . . . . . . . 9 |- ( invr ` S ) = ( invr ` S )
46		eqid 2238	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
47	1, 45, 46	ringinvcl 14405	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
48	8, 47	sylan 283	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
49	38, 48	sseldd 3249	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` R ) )
50	28, 29, 32, 33, 44, 49	dvdsrmuld 14376	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) )
51	1, 45	unitinvcl 14403	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
52	8, 51	sylan 283	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
53		eqid 2238	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
54		eqid 2238	. . . . . . . 8 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
55	26, 53, 25, 54	opprmulg 14349	. . . . . . 7 |- ( ( R e. Ring /\ ( ( invr ` S ) ` x ) e. V /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) )
56	24, 52, 42, 55	syl3anc 1278	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) )
57		eqid 2238	. . . . . . . . 9 |- ( .r ` S ) = ( .r ` S )
58		eqid 2238	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
59	1, 45, 57, 58	unitrinv 14407	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) = ( 1r ` S ) )
60	8, 59	sylan 283	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) = ( 1r ` S ) )
61	7, 53	ressmulrg 13476	. . . . . . . . . 10 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
62	23, 61	mpdan 425	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
63	62	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` R ) = ( .r ` S ) )
64	63	oveqd 6092	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) )
65	60, 64, 15	3eqtr4d 2281	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( 1r ` R ) )
66	56, 65	eqtrd 2271	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
67	50, 66	breqtrd 4151	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
68		subrguss.2	. . . . . . 7 |- U = (Unit ` R )
69	68	a1i 9	. . . . . 6 |- ( A e. (SubRing ` R ) -> U = (Unit ` R ) )
70		eqidd 2239	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` R ) )
71		eqidd 2239	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` R ) = ( ||r ` R ) )
72		eqidd 2239	. . . . . 6 |- ( A e. (SubRing ` R ) -> (oppr ` R ) = (oppr ` R ) )
73		eqidd 2239	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
74		ringsrg 14325	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
75	23, 74	syl 14	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. SRing )
76	69, 70, 71, 72, 73, 75	isunitd 14386	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
77	76	adantr 276	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
78	22, 67, 77	mpbir2and 957	. . 3 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. U )
79	78	ex 115	. 2 |- ( A e. (SubRing ` R ) -> ( x e. V -> x e. U ) )
80	79	ssrdv 3254	1 |- ( A e. (SubRing ` R ) -> V C_ U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   .rcmulr 13409   1rcur 14237  SRingcsrg 14241   Ringcrg 14274  opp_rcoppr 14345   ||_rcdsr 14365  Unitcui 14366   invrcinvr 14400  SubRingcsubrg 14498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-ur 14238  df-srg 14242  df-ring 14276  df-oppr 14346  df-dvdsr 14368  df-unit 14369  df-invr 14401  df-subrg 14500

This theorem is referenced by:  subrginv  14518  subrgdv  14519  subrgunit  14520  subrgugrp  14521