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Theorem subrginv 13296
Description: A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrginv.1  |-  S  =  ( Rs  A )
subrginv.2  |-  I  =  ( invr `  R
)
subrginv.3  |-  U  =  (Unit `  S )
subrginv.4  |-  J  =  ( invr `  S
)
Assertion
Ref Expression
subrginv  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  =  ( J `  X ) )

Proof of Theorem subrginv
StepHypRef Expression
1 subrgrcl 13285 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
21adantr 276 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  R  e.  Ring )
3 subrginv.1 . . . . . . . 8  |-  S  =  ( Rs  A )
43subrgbas 13289 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
5 eqid 2177 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
65subrgss 13281 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
74, 6eqsstrrd 3192 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
87adantr 276 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( Base `  S )  C_  ( Base `  R )
)
93subrgring 13283 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
10 subrginv.3 . . . . . . 7  |-  U  =  (Unit `  S )
11 subrginv.4 . . . . . . 7  |-  J  =  ( invr `  S
)
12 eqid 2177 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
1310, 11, 12ringinvcl 13225 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
149, 13sylan 283 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
158, 14sseldd 3156 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  R
) )
16 eqidd 2178 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( Base `  S )  =  ( Base `  S
) )
1710a1i 9 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  U  =  (Unit `  S )
)
189adantr 276 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  S  e.  Ring )
19 ringsrg 13155 . . . . . . 7  |-  ( S  e.  Ring  ->  S  e. SRing
)
2018, 19syl 14 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  S  e. SRing )
21 simpr 110 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  U )
2216, 17, 20, 21unitcld 13208 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  S
) )
238, 22sseldd 3156 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  R
) )
24 eqid 2177 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
253, 24, 10subrguss 13295 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
2625sselda 3155 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  (Unit `  R )
)
27 subrginv.2 . . . . . 6  |-  I  =  ( invr `  R
)
2824, 27, 5ringinvcl 13225 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( I `  X )  e.  (
Base `  R )
)
291, 26, 28syl2an2r 595 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  e.  ( Base `  R
) )
30 eqid 2177 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
315, 30ringass 13130 . . . 4  |-  ( ( R  e.  Ring  /\  (
( J `  X
)  e.  ( Base `  R )  /\  X  e.  ( Base `  R
)  /\  ( I `  X )  e.  (
Base `  R )
) )  ->  (
( ( J `  X ) ( .r
`  R ) X ) ( .r `  R ) ( I `
 X ) )  =  ( ( J `
 X ) ( .r `  R ) ( X ( .r
`  R ) ( I `  X ) ) ) )
322, 15, 23, 29, 31syl13anc 1240 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( ( J `  X ) ( .r
`  R ) X ) ( .r `  R ) ( I `
 X ) )  =  ( ( J `
 X ) ( .r `  R ) ( X ( .r
`  R ) ( I `  X ) ) ) )
33 eqid 2177 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
34 eqid 2177 . . . . . . 7  |-  ( 1r
`  S )  =  ( 1r `  S
)
3510, 11, 33, 34unitlinv 13226 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
369, 35sylan 283 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
373, 30ressmulrg 12595 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
381, 37mpdan 421 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3938adantr 276 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( .r `  R )  =  ( .r `  S
) )
4039oveqd 5889 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( ( J `
 X ) ( .r `  S ) X ) )
41 eqid 2177 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
423, 41subrg1 13290 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
4342adantr 276 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( 1r `  R )  =  ( 1r `  S
) )
4436, 40, 433eqtr4d 2220 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
4544oveq1d 5887 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( ( J `  X ) ( .r
`  R ) X ) ( .r `  R ) ( I `
 X ) )  =  ( ( 1r
`  R ) ( .r `  R ) ( I `  X
) ) )
4624, 27, 30, 41unitrinv 13227 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( X
( .r `  R
) ( I `  X ) )  =  ( 1r `  R
) )
471, 26, 46syl2an2r 595 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
4847oveq2d 5888 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) ( X ( .r `  R
) ( I `  X ) ) )  =  ( ( J `
 X ) ( .r `  R ) ( 1r `  R
) ) )
4932, 45, 483eqtr3d 2218 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( ( J `
 X ) ( .r `  R ) ( 1r `  R
) ) )
505, 30, 41ringlidm 13137 . . 3  |-  ( ( R  e.  Ring  /\  (
I `  X )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
511, 29, 50syl2an2r 595 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
525, 30, 41ringridm 13138 . . 3  |-  ( ( R  e.  Ring  /\  ( J `  X )  e.  ( Base `  R
) )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
531, 15, 52syl2an2r 595 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
5449, 51, 533eqtr3d 2218 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  =  ( J `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   .rcmulr 12529   1rcur 13073  SRingcsrg 13077   Ringcrg 13110  Unitcui 13187   invrcinvr 13220  SubRingcsubrg 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-oppr 13171  df-dvdsr 13189  df-unit 13190  df-invr 13221  df-subrg 13278
This theorem is referenced by:  subrgdv  13297  subrgunit  13298  subrgugrp  13299
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