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Theorem subrguss 15876
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 448 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  V )
2 subrguss.3 . . . . . . . . 9  |-  V  =  (Unit `  S )
3 eqid 2436 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
4 eqid 2436 . . . . . . . . 9  |-  ( ||r `  S
)  =  ( ||r `  S
)
5 eqid 2436 . . . . . . . . 9  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2436 . . . . . . . . 9  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
72, 3, 4, 5, 6isunit 15755 . . . . . . . 8  |-  ( x  e.  V  <->  ( x
( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
81, 7sylib 189 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
98simpld 446 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  S ) )
10 subrguss.1 . . . . . . . 8  |-  S  =  ( Rs  A )
11 eqid 2436 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
1210, 11subrg1 15871 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
1312adantr 452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( 1r `  R )  =  ( 1r `  S
) )
149, 13breqtrrd 4231 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  R ) )
15 eqid 2436 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
1610, 15, 4subrgdvds 15875 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  S
)  C_  ( ||r `  R
) )
1716adantr 452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( ||r `  S )  C_  ( ||r `  R ) )
1817ssbrd 4246 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  R )  ->  x ( ||r `  R
) ( 1r `  R ) ) )
1914, 18mpd 15 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 R ) ( 1r `  R ) )
2010subrgbas 15870 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
2120adantr 452 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  =  ( Base `  S
) )
22 eqid 2436 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2322subrgss 15862 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
2423adantr 452 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  C_  ( Base `  R
) )
2521, 24eqsstr3d 3376 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  C_  ( Base `  R )
)
26 eqid 2436 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
2726, 2unitcl 15757 . . . . . . . 8  |-  ( x  e.  V  ->  x  e.  ( Base `  S
) )
2827adantl 453 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  S
) )
2925, 28sseldd 3342 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3010subrgrng 15864 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
31 eqid 2436 . . . . . . . . 9  |-  ( invr `  S )  =  (
invr `  S )
322, 31, 26rnginvcl 15774 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3330, 32sylan 458 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3425, 33sseldd 3342 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  R )
)
35 eqid 2436 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
3635, 22opprbas 15727 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
37 eqid 2436 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
38 eqid 2436 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3936, 37, 38dvdsrmul 15746 . . . . . 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 643 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
41 eqid 2436 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4222, 41, 35, 38opprmul 15724 . . . . . 6  |-  ( ( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) ( ( invr `  S
) `  x )
)
43 eqid 2436 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
442, 31, 43, 3unitrinv 15776 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4530, 44sylan 458 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4610, 41ressmulr 13575 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
4746adantr 452 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  R )  =  ( .r `  S
) )
4847oveqd 6091 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( x ( .r
`  S ) ( ( invr `  S
) `  x )
) )
4945, 48, 133eqtr4d 2478 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( 1r `  R
) )
5042, 49syl5eq 2480 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )
5140, 50breqtrd 4229 . . . 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 15755 . . . 4  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
5419, 51, 53sylanbrc 646 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  U )
5554ex 424 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  V  ->  x  e.  U ) )
5655ssrdv 3347 1  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3313   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   ↾s cress 13463   .rcmulr 13523   Ringcrg 15653   1rcur 15655  opprcoppr 15720   ||rcdsr 15736  Unitcui 15737   invrcinvr 15769  SubRingcsubrg 15857
This theorem is referenced by:  subrginv  15877  subrgdv  15878  subrgunit  15879  subrgugrp  15880  issubdrg  15886  zrngunit  16758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-tpos 6472  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-0g 13720  df-mnd 14683  df-grp 14805  df-minusg 14806  df-subg 14934  df-mgp 15642  df-rng 15656  df-ur 15658  df-oppr 15721  df-dvdsr 15739  df-unit 15740  df-invr 15770  df-subrg 15859
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