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Theorem subrguss 15576
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 simpr 447 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  V )
2 subrguss.3 . . . . . . . . 9  |-  V  =  (Unit `  S )
3 eqid 2296 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
4 eqid 2296 . . . . . . . . 9  |-  ( ||r `  S
)  =  ( ||r `  S
)
5 eqid 2296 . . . . . . . . 9  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2296 . . . . . . . . 9  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
72, 3, 4, 5, 6isunit 15455 . . . . . . . 8  |-  ( x  e.  V  <->  ( x
( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
81, 7sylib 188 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
98simpld 445 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  S ) )
10 subrguss.1 . . . . . . . 8  |-  S  =  ( Rs  A )
11 eqid 2296 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
1210, 11subrg1 15571 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
1312adantr 451 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( 1r `  R )  =  ( 1r `  S
) )
149, 13breqtrrd 4065 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  R ) )
15 eqid 2296 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
1610, 15, 4subrgdvds 15575 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  S
)  C_  ( ||r `  R
) )
1716adantr 451 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( ||r `  S )  C_  ( ||r `  R ) )
1817ssbrd 4080 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  R )  ->  x ( ||r `  R
) ( 1r `  R ) ) )
1914, 18mpd 14 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 R ) ( 1r `  R ) )
2010subrgbas 15570 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
2120adantr 451 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  =  ( Base `  S
) )
22 eqid 2296 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2322subrgss 15562 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
2423adantr 451 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  C_  ( Base `  R
) )
2521, 24eqsstr3d 3226 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  C_  ( Base `  R )
)
26 eqid 2296 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
2726, 2unitcl 15457 . . . . . . . 8  |-  ( x  e.  V  ->  x  e.  ( Base `  S
) )
2827adantl 452 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  S
) )
2925, 28sseldd 3194 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3010subrgrng 15564 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
31 eqid 2296 . . . . . . . . 9  |-  ( invr `  S )  =  (
invr `  S )
322, 31, 26rnginvcl 15474 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3330, 32sylan 457 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3425, 33sseldd 3194 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  R )
)
35 eqid 2296 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
3635, 22opprbas 15427 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
37 eqid 2296 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
38 eqid 2296 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3936, 37, 38dvdsrmul 15446 . . . . . 6  |-  ( ( x  e.  ( Base `  R )  /\  (
( invr `  S ) `  x )  e.  (
Base `  R )
)  ->  x ( ||r `  (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
4029, 34, 39syl2anc 642 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
41 eqid 2296 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4222, 41, 35, 38opprmul 15424 . . . . . 6  |-  ( ( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) ( ( invr `  S
) `  x )
)
43 eqid 2296 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
442, 31, 43, 3unitrinv 15476 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4530, 44sylan 457 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4610, 41ressmulr 13277 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
4746adantr 451 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  R )  =  ( .r `  S
) )
4847oveqd 5891 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( x ( .r
`  S ) ( ( invr `  S
) `  x )
) )
4945, 48, 133eqtr4d 2338 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( 1r `  R
) )
5042, 49syl5eq 2340 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )
5140, 50breqtrd 4063 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
52 subrguss.2 . . . . 5  |-  U  =  (Unit `  R )
5352, 11, 15, 35, 37isunit 15455 . . . 4  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
5419, 51, 53sylanbrc 645 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  U )
5554ex 423 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  V  ->  x  e.  U ) )
5655ssrdv 3198 1  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   .rcmulr 13225   Ringcrg 15353   1rcur 15355  opprcoppr 15420   ||rcdsr 15436  Unitcui 15437   invrcinvr 15469  SubRingcsubrg 15557
This theorem is referenced by:  subrginv  15577  subrgdv  15578  subrgunit  15579  subrgugrp  15580  issubdrg  15586  zrngunit  16454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-subrg 15559
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