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Theorem subrguss 13295
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  =  ( Rs  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.
StepHypRef Expression
1 subrguss.3 . . . . . . . . 9  |-  V  =  (Unit `  S )
21a1i 9 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  V  =  (Unit `  S ) )
3 eqidd 2178 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  S )  =  ( 1r `  S ) )
4 eqidd 2178 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  S
)  =  ( ||r `  S
) )
5 eqidd 2178 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  (oppr
`  S )  =  (oppr
`  S ) )
6 eqidd 2178 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) ) )
7 subrguss.1 . . . . . . . . . 10  |-  S  =  ( Rs  A )
87subrgring 13283 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
9 ringsrg 13155 . . . . . . . . 9  |-  ( S  e.  Ring  ->  S  e. SRing
)
108, 9syl 14 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  S  e. SRing )
112, 3, 4, 5, 6, 10isunitd 13206 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  V  <->  ( x (
||r `  S ) ( 1r
`  S )  /\  x ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) ) ) )
1211simprbda 383 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  S ) )
13 eqid 2177 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
147, 13subrg1 13290 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
1514adantr 276 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( 1r `  R )  =  ( 1r `  S
) )
1612, 15breqtrrd 4030 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  R ) )
17 eqid 2177 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
18 eqid 2177 . . . . . . . 8  |-  ( ||r `  S
)  =  ( ||r `  S
)
197, 17, 18subrgdvds 13294 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  S
)  C_  ( ||r `  R
) )
2019adantr 276 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( ||r `  S )  C_  ( ||r `  R ) )
2120ssbrd 4045 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  R )  ->  x ( ||r `  R
) ( 1r `  R ) ) )
2216, 21mpd 13 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 R ) ( 1r `  R ) )
23 subrgrcl 13285 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
2423adantr 276 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  R  e.  Ring )
25 eqid 2177 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
26 eqid 2177 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
2725, 26opprbasg 13178 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  (oppr
`  R ) ) )
2824, 27syl 14 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  R )  =  ( Base `  (oppr `  R
) ) )
29 eqidd 2178 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( ||r `  (oppr
`  R ) )  =  ( ||r `
 (oppr
`  R ) ) )
3025opprring 13180 . . . . . . 7  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
31 ringsrg 13155 . . . . . . 7  |-  ( (oppr `  R )  e.  Ring  -> 
(oppr `  R )  e. SRing )
3224, 30, 313syl 17 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (oppr `  R
)  e. SRing )
33 eqidd 2178 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) ) )
347subrgbas 13289 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
3534adantr 276 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  =  ( Base `  S
) )
3626subrgss 13281 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
3736adantr 276 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  C_  ( Base `  R
) )
3835, 37eqsstrrd 3192 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  C_  ( Base `  R )
)
39 eqidd 2178 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  =  ( Base `  S
) )
401a1i 9 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  V  =  (Unit `  S )
)
4110adantr 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 )
4339, 40, 41, 42unitcld 13208 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  S
) )
4438, 43sseldd 3156 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
45 eqid 2177 . . . . . . . . 9  |-  ( invr `  S )  =  (
invr `  S )
46 eqid 2177 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
471, 45, 46ringinvcl 13225 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
488, 47sylan 283 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
4938, 48sseldd 3156 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  R )
)
5028, 29, 32, 33, 44, 49dvdsrmuld 13196 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
511, 45unitinvcl 13223 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
528, 51sylan 283 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
53 eqid 2177 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
54 eqid 2177 . . . . . . . 8  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5526, 53, 25, 54opprmulg 13174 . . . . . . 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 ) ) )
5624, 52, 42, 55syl3anc 1238 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) ( ( invr `  S
) `  x )
) )
57 eqid 2177 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
58 eqid 2177 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
591, 45, 57, 58unitrinv 13227 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
608, 59sylan 283 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
617, 53ressmulrg 12595 . . . . . . . . . 10  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
6223, 61mpdan 421 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
6362adantr 276 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  R )  =  ( .r `  S
) )
6463oveqd 5889 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( x ( .r
`  S ) ( ( invr `  S
) `  x )
) )
6560, 64, 153eqtr4d 2220 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( 1r `  R
) )
6656, 65eqtrd 2210 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )
6750, 66breqtrd 4028 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
68 subrguss.2 . . . . . . 7  |-  U  =  (Unit `  R )
6968a1i 9 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  U  =  (Unit `  R ) )
70 eqidd 2178 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  R ) )
71 eqidd 2178 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  R
)  =  ( ||r `  R
) )
72 eqidd 2178 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  (oppr
`  R )  =  (oppr
`  R ) )
73 eqidd 2178 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) ) )
74 ringsrg 13155 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. SRing
)
7523, 74syl 14 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  R  e. SRing )
7669, 70, 71, 72, 73, 75isunitd 13206 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  U  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) ) )
7776adantr 276 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
7822, 67, 77mpbir2and 944 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  U )
7978ex 115 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  V  ->  x  e.  U ) )
8079ssrdv 3161 1  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    C_ wss 3129   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   .rcmulr 12529   1rcur 13073  SRingcsrg 13077   Ringcrg 13110  opprcoppr 13170   ||rcdsr 13186  Unitcui 13187   invrcinvr 13220  SubRingcsubrg 13276
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-oppr 13171  df-dvdsr 13189  df-unit 13190  df-invr 13221  df-subrg 13278
This theorem is referenced by:  subrginv  13296  subrgdv  13297  subrgunit  13298  subrgugrp  13299
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