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Theorem subrguss 17256
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 461 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  V )
2 subrguss.3 . . . . . . . . 9  |-  V  =  (Unit `  S )
3 eqid 2467 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
4 eqid 2467 . . . . . . . . 9  |-  ( ||r `  S
)  =  ( ||r `  S
)
5 eqid 2467 . . . . . . . . 9  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2467 . . . . . . . . 9  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
72, 3, 4, 5, 6isunit 17119 . . . . . . . 8  |-  ( x  e.  V  <->  ( x
( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
81, 7sylib 196 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
98simpld 459 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  S ) )
10 subrguss.1 . . . . . . . 8  |-  S  =  ( Rs  A )
11 eqid 2467 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
1210, 11subrg1 17251 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
1312adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( 1r `  R )  =  ( 1r `  S
) )
149, 13breqtrrd 4473 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  R ) )
15 eqid 2467 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
1610, 15, 4subrgdvds 17255 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  S
)  C_  ( ||r `  R
) )
1716adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( ||r `  S )  C_  ( ||r `  R ) )
1817ssbrd 4488 . . . . 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 17250 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
2120adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  =  ( Base `  S
) )
22 eqid 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2322subrgss 17242 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
2423adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  C_  ( Base `  R
) )
2521, 24eqsstr3d 3539 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  C_  ( Base `  R )
)
26 eqid 2467 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
2726, 2unitcl 17121 . . . . . . . 8  |-  ( x  e.  V  ->  x  e.  ( Base `  S
) )
2827adantl 466 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  S
) )
2925, 28sseldd 3505 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3010subrgrng 17244 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
31 eqid 2467 . . . . . . . . 9  |-  ( invr `  S )  =  (
invr `  S )
322, 31, 26rnginvcl 17138 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3330, 32sylan 471 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3425, 33sseldd 3505 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  R )
)
35 eqid 2467 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
3635, 22opprbas 17091 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
37 eqid 2467 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
38 eqid 2467 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3936, 37, 38dvdsrmul 17110 . . . . . 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 661 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
41 eqid 2467 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4222, 41, 35, 38opprmul 17088 . . . . . 6  |-  ( ( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) ( ( invr `  S
) `  x )
)
43 eqid 2467 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
442, 31, 43, 3unitrinv 17140 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4530, 44sylan 471 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4610, 41ressmulr 14611 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
4746adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  R )  =  ( .r `  S
) )
4847oveqd 6302 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( x ( .r
`  S ) ( ( invr `  S
) `  x )
) )
4945, 48, 133eqtr4d 2518 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( 1r `  R
) )
5042, 49syl5eq 2520 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )
5140, 50breqtrd 4471 . . . 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 17119 . . . 4  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
5419, 51, 53sylanbrc 664 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  U )
5554ex 434 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  V  ->  x  e.  U ) )
5655ssrdv 3510 1  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   ↾s cress 14494   .rcmulr 14559   1rcur 16967   Ringcrg 17012  opprcoppr 17084   ||rcdsr 17100  Unitcui 17101   invrcinvr 17133  SubRingcsubrg 17237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-subg 16012  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-subrg 17239
This theorem is referenced by:  subrginv  17257  subrgdv  17258  subrgunit  17259  subrgugrp  17260  issubdrg  17266  zringunit  18327  zrngunit  18328
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