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Theorem subrginv 15577
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 15566 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
21adantr 451 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  R  e.  Ring )
3 subrginv.1 . . . . . . . 8  |-  S  =  ( Rs  A )
43subrgbas 15570 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
5 eqid 2296 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
65subrgss 15562 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
74, 6eqsstr3d 3226 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
87adantr 451 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( Base `  S )  C_  ( Base `  R )
)
93subrgrng 15564 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
10 subrginv.3 . . . . . . 7  |-  U  =  (Unit `  S )
11 subrginv.4 . . . . . . 7  |-  J  =  ( invr `  S
)
12 eqid 2296 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
1310, 11, 12rnginvcl 15474 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
149, 13sylan 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
158, 14sseldd 3194 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  R
) )
1612, 10unitcl 15457 . . . . . 6  |-  ( X  e.  U  ->  X  e.  ( Base `  S
) )
1716adantl 452 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  S
) )
188, 17sseldd 3194 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  R
) )
19 eqid 2296 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
203, 19, 10subrguss 15576 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
2120sselda 3193 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  (Unit `  R )
)
22 subrginv.2 . . . . . . 7  |-  I  =  ( invr `  R
)
2319, 22, 5rnginvcl 15474 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( I `  X )  e.  (
Base `  R )
)
241, 23sylan 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  (Unit `  R ) )  ->  ( I `  X )  e.  (
Base `  R )
)
2521, 24syldan 456 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  e.  ( Base `  R
) )
26 eqid 2296 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
275, 26rngass 15373 . . . 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 ) ) ) )
282, 15, 18, 25, 27syl13anc 1184 . . 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 ) ) ) )
29 eqid 2296 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
30 eqid 2296 . . . . . . 7  |-  ( 1r
`  S )  =  ( 1r `  S
)
3110, 11, 29, 30unitlinv 15475 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
329, 31sylan 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
333, 26ressmulr 13277 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3433adantr 451 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( .r `  R )  =  ( .r `  S
) )
3534oveqd 5891 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( ( J `
 X ) ( .r `  S ) X ) )
36 eqid 2296 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
373, 36subrg1 15571 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
3837adantr 451 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( 1r `  R )  =  ( 1r `  S
) )
3932, 35, 383eqtr4d 2338 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
4039oveq1d 5889 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( ( J `  X ) ( .r
`  R ) X ) ( .r `  R ) ( I `
 X ) )  =  ( ( 1r
`  R ) ( .r `  R ) ( I `  X
) ) )
4119, 22, 26, 36unitrinv 15476 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( X
( .r `  R
) ( I `  X ) )  =  ( 1r `  R
) )
421, 41sylan 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  (Unit `  R ) )  ->  ( X ( .r `  R ) ( I `  X
) )  =  ( 1r `  R ) )
4321, 42syldan 456 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
4443oveq2d 5890 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) ( X ( .r `  R
) ( I `  X ) ) )  =  ( ( J `
 X ) ( .r `  R ) ( 1r `  R
) ) )
4528, 40, 443eqtr3d 2336 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( ( J `
 X ) ( .r `  R ) ( 1r `  R
) ) )
465, 26, 36rnglidm 15380 . . 3  |-  ( ( R  e.  Ring  /\  (
I `  X )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
472, 25, 46syl2anc 642 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
485, 26, 36rngridm 15381 . . 3  |-  ( ( R  e.  Ring  /\  ( J `  X )  e.  ( Base `  R
) )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
492, 15, 48syl2anc 642 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
5045, 47, 493eqtr3d 2336 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  =  ( J `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   .rcmulr 13225   Ringcrg 15353   1rcur 15355  Unitcui 15437   invrcinvr 15469  SubRingcsubrg 15557
This theorem is referenced by:  subrgdv  15578  subrgunit  15579  subrgugrp  15580  issubdrg  15586  gzrngunit  16453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-subrg 15559
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