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Theorem subrguss 17006
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 2454 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
4 eqid 2454 . . . . . . . . 9  |-  ( ||r `  S
)  =  ( ||r `  S
)
5 eqid 2454 . . . . . . . . 9  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2454 . . . . . . . . 9  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
72, 3, 4, 5, 6isunit 16875 . . . . . . . 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 2454 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
1210, 11subrg1 17001 . . . . . . 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 4429 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  R ) )
15 eqid 2454 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
1610, 15, 4subrgdvds 17005 . . . . . . 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 4444 . . . . 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 17000 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
2120adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  =  ( Base `  S
) )
22 eqid 2454 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2322subrgss 16992 . . . . . . . . 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 3502 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  C_  ( Base `  R )
)
26 eqid 2454 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
2726, 2unitcl 16877 . . . . . . . 8  |-  ( x  e.  V  ->  x  e.  ( Base `  S
) )
2827adantl 466 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  S
) )
2925, 28sseldd 3468 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3010subrgrng 16994 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
31 eqid 2454 . . . . . . . . 9  |-  ( invr `  S )  =  (
invr `  S )
322, 31, 26rnginvcl 16894 . . . . . . . 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 3468 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  R )
)
35 eqid 2454 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
3635, 22opprbas 16847 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
37 eqid 2454 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
38 eqid 2454 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3936, 37, 38dvdsrmul 16866 . . . . . 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 2454 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4222, 41, 35, 38opprmul 16844 . . . . . 6  |-  ( ( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) ( ( invr `  S
) `  x )
)
43 eqid 2454 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
442, 31, 43, 3unitrinv 16896 . . . . . . . 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 14413 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
4746adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  R )  =  ( .r `  S
) )
4847oveqd 6220 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( x ( .r
`  S ) ( ( invr `  S
) `  x )
) )
4945, 48, 133eqtr4d 2505 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( 1r `  R
) )
5042, 49syl5eq 2507 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )
5140, 50breqtrd 4427 . . . 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 16875 . . . 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 3473 1  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3439   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   ↾s cress 14296   .rcmulr 14361   1rcur 16728   Ringcrg 16771  opprcoppr 16840   ||rcdsr 16856  Unitcui 16857   invrcinvr 16889  SubRingcsubrg 16987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-tpos 6858  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668  df-subg 15800  df-mgp 16717  df-ur 16729  df-rng 16773  df-oppr 16841  df-dvdsr 16859  df-unit 16860  df-invr 16890  df-subrg 16989
This theorem is referenced by:  subrginv  17007  subrgdv  17008  subrgunit  17009  subrgugrp  17010  issubdrg  17016  zringunit  18042  zrngunit  18043
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