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Theorem subrgunit 15886
Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgugrp.1  |-  S  =  ( Rs  A )
subrgugrp.2  |-  U  =  (Unit `  R )
subrgugrp.3  |-  V  =  (Unit `  S )
subrgunit.4  |-  I  =  ( invr `  R
)
Assertion
Ref Expression
subrgunit  |-  ( A  e.  (SubRing `  R
)  ->  ( X  e.  V  <->  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) ) )

Proof of Theorem subrgunit
StepHypRef Expression
1 subrgugrp.1 . . . . 5  |-  S  =  ( Rs  A )
2 subrgugrp.2 . . . . 5  |-  U  =  (Unit `  R )
3 subrgugrp.3 . . . . 5  |-  V  =  (Unit `  S )
41, 2, 3subrguss 15883 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
54sselda 3348 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  U )
6 eqid 2436 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
76, 3unitcl 15764 . . . . 5  |-  ( X  e.  V  ->  X  e.  ( Base `  S
) )
87adantl 453 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  ( Base `  S
) )
91subrgbas 15877 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
109adantr 452 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  A  =  ( Base `  S
) )
118, 10eleqtrrd 2513 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  A )
121subrgrng 15871 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
13 eqid 2436 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
143, 13, 6rnginvcl 15781 . . . . 5  |-  ( ( S  e.  Ring  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
1512, 14sylan 458 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
16 subrgunit.4 . . . . 5  |-  I  =  ( invr `  R
)
171, 16, 3, 13subrginv 15884 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  =  ( ( invr `  S ) `  X
) )
1815, 17, 103eltr4d 2517 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  e.  A )
195, 11, 183jca 1134 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  ( X  e.  U  /\  X  e.  A  /\  ( I `  X
)  e.  A ) )
20 simpr2 964 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  A )
219adantr 452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  A  =  ( Base `  S ) )
2220, 21eleqtrd 2512 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  ( Base `  S ) )
23 simpr3 965 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  A )
2423, 21eleqtrd 2512 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  ( Base `  S ) )
25 eqid 2436 . . . . . 6  |-  ( ||r `  S
)  =  ( ||r `  S
)
26 eqid 2436 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
276, 25, 26dvdsrmul 15753 . . . . 5  |-  ( ( X  e.  ( Base `  S )  /\  (
I `  X )  e.  ( Base `  S
) )  ->  X
( ||r `
 S ) ( ( I `  X
) ( .r `  S ) X ) )
2822, 24, 27syl2anc 643 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( ( I `  X
) ( .r `  S ) X ) )
29 subrgrcl 15873 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
3029adantr 452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  R  e.  Ring )
31 simpr1 963 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  U )
32 eqid 2436 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
33 eqid 2436 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
342, 16, 32, 33unitlinv 15782 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( I `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
3530, 31, 34syl2anc 643 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  R ) X )  =  ( 1r
`  R ) )
361, 32ressmulr 13582 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3736adantr 452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( .r `  R
)  =  ( .r
`  S ) )
3837oveqd 6098 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  R ) X )  =  ( ( I `  X ) ( .r `  S
) X ) )
391, 33subrg1 15878 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
4039adantr 452 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( 1r `  R
)  =  ( 1r
`  S ) )
4135, 38, 403eqtr3d 2476 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  S ) X )  =  ( 1r
`  S ) )
4228, 41breqtrd 4236 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( 1r `  S ) )
43 eqid 2436 . . . . . . 7  |-  (oppr `  S
)  =  (oppr `  S
)
4443, 6opprbas 15734 . . . . . 6  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
45 eqid 2436 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
46 eqid 2436 . . . . . 6  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
4744, 45, 46dvdsrmul 15753 . . . . 5  |-  ( ( X  e.  ( Base `  S )  /\  (
I `  X )  e.  ( Base `  S
) )  ->  X
( ||r `
 (oppr
`  S ) ) ( ( I `  X ) ( .r
`  (oppr
`  S ) ) X ) )
4822, 24, 47syl2anc 643 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 (oppr
`  S ) ) ( ( I `  X ) ( .r
`  (oppr
`  S ) ) X ) )
496, 26, 43, 46opprmul 15731 . . . . 5  |-  ( ( I `  X ) ( .r `  (oppr `  S
) ) X )  =  ( X ( .r `  S ) ( I `  X
) )
502, 16, 32, 33unitrinv 15783 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
5130, 31, 50syl2anc 643 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  R ) ( I `  X ) )  =  ( 1r
`  R ) )
5237oveqd 6098 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  R ) ( I `  X ) )  =  ( X ( .r `  S
) ( I `  X ) ) )
5351, 52, 403eqtr3d 2476 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  S ) ( I `  X ) )  =  ( 1r
`  S ) )
5449, 53syl5eq 2480 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  (oppr
`  S ) ) X )  =  ( 1r `  S ) )
5548, 54breqtrd 4236 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) )
56 eqid 2436 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
573, 56, 25, 43, 45isunit 15762 . . 3  |-  ( X  e.  V  <->  ( X
( ||r `
 S ) ( 1r `  S )  /\  X ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
5842, 55, 57sylanbrc 646 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  V )
5919, 58impbida 806 1  |-  ( A  e.  (SubRing `  R
)  ->  ( X  e.  V  <->  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   .rcmulr 13530   Ringcrg 15660   1rcur 15662  opprcoppr 15727   ||rcdsr 15743  Unitcui 15744   invrcinvr 15776  SubRingcsubrg 15864
This theorem is referenced by:  issubdrg  15893  gzrngunit  16764  zrngunit  16765  cphreccllem  19141
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 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-subrg 15866
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