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Theorem subrginv 18036
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 18025 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
21adantr 467 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  R  e.  Ring )
3 subrginv.1 . . . . . . . 8  |-  S  =  ( Rs  A )
43subrgbas 18029 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
5 eqid 2453 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
65subrgss 18021 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
74, 6eqsstr3d 3469 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
87adantr 467 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( Base `  S )  C_  ( Base `  R )
)
93subrgring 18023 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
10 subrginv.3 . . . . . . 7  |-  U  =  (Unit `  S )
11 subrginv.4 . . . . . . 7  |-  J  =  ( invr `  S
)
12 eqid 2453 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
1310, 11, 12ringinvcl 17916 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
149, 13sylan 474 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
158, 14sseldd 3435 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  R
) )
1612, 10unitcl 17899 . . . . . 6  |-  ( X  e.  U  ->  X  e.  ( Base `  S
) )
1716adantl 468 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  S
) )
188, 17sseldd 3435 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  R
) )
19 eqid 2453 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
203, 19, 10subrguss 18035 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
2120sselda 3434 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  (Unit `  R )
)
22 subrginv.2 . . . . . . 7  |-  I  =  ( invr `  R
)
2319, 22, 5ringinvcl 17916 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( I `  X )  e.  (
Base `  R )
)
241, 23sylan 474 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  (Unit `  R ) )  ->  ( I `  X )  e.  (
Base `  R )
)
2521, 24syldan 473 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  e.  ( Base `  R
) )
26 eqid 2453 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
275, 26ringass 17809 . . . 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 1271 . . 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 2453 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
30 eqid 2453 . . . . . . 7  |-  ( 1r
`  S )  =  ( 1r `  S
)
3110, 11, 29, 30unitlinv 17917 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
329, 31sylan 474 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
333, 26ressmulr 15262 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3433adantr 467 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( .r `  R )  =  ( .r `  S
) )
3534oveqd 6312 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( ( J `
 X ) ( .r `  S ) X ) )
36 eqid 2453 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
373, 36subrg1 18030 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
3837adantr 467 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( 1r `  R )  =  ( 1r `  S
) )
3932, 35, 383eqtr4d 2497 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
4039oveq1d 6310 . . 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 17918 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( X
( .r `  R
) ( I `  X ) )  =  ( 1r `  R
) )
421, 41sylan 474 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  (Unit `  R ) )  ->  ( X ( .r `  R ) ( I `  X
) )  =  ( 1r `  R ) )
4321, 42syldan 473 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
4443oveq2d 6311 . . 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 2495 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( ( J `
 X ) ( .r `  R ) ( 1r `  R
) ) )
465, 26, 36ringlidm 17816 . . 3  |-  ( ( R  e.  Ring  /\  (
I `  X )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
472, 25, 46syl2anc 667 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
485, 26, 36ringridm 17817 . . 3  |-  ( ( R  e.  Ring  /\  ( J `  X )  e.  ( Base `  R
) )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
492, 15, 48syl2anc 667 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
5045, 47, 493eqtr3d 2495 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  =  ( J `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889    C_ wss 3406   ` cfv 5585  (class class class)co 6295   Basecbs 15133   ↾s cress 15134   .rcmulr 15203   1rcur 17747   Ringcrg 17792  Unitcui 17879   invrcinvr 17911  SubRingcsubrg 18016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-grp 16685  df-minusg 16686  df-subg 16826  df-mgp 17736  df-ur 17748  df-ring 17794  df-oppr 17863  df-dvdsr 17881  df-unit 17882  df-invr 17912  df-subrg 18018
This theorem is referenced by:  subrgdv  18037  subrgunit  18038  subrgugrp  18039  issubdrg  18045  gzrngunit  19045
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