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Theorem subrginv 15884
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 15873 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
21adantr 452 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  R  e.  Ring )
3 subrginv.1 . . . . . . . 8  |-  S  =  ( Rs  A )
43subrgbas 15877 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
5 eqid 2436 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
65subrgss 15869 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
74, 6eqsstr3d 3383 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
87adantr 452 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( Base `  S )  C_  ( Base `  R )
)
93subrgrng 15871 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
10 subrginv.3 . . . . . . 7  |-  U  =  (Unit `  S )
11 subrginv.4 . . . . . . 7  |-  J  =  ( invr `  S
)
12 eqid 2436 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
1310, 11, 12rnginvcl 15781 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
149, 13sylan 458 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  S
) )
158, 14sseldd 3349 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( J `  X )  e.  ( Base `  R
) )
1612, 10unitcl 15764 . . . . . 6  |-  ( X  e.  U  ->  X  e.  ( Base `  S
) )
1716adantl 453 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  S
) )
188, 17sseldd 3349 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  ( Base `  R
) )
19 eqid 2436 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
203, 19, 10subrguss 15883 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
2120sselda 3348 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  X  e.  (Unit `  R )
)
22 subrginv.2 . . . . . . 7  |-  I  =  ( invr `  R
)
2319, 22, 5rnginvcl 15781 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( I `  X )  e.  (
Base `  R )
)
241, 23sylan 458 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  (Unit `  R ) )  ->  ( I `  X )  e.  (
Base `  R )
)
2521, 24syldan 457 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  e.  ( Base `  R
) )
26 eqid 2436 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
275, 26rngass 15680 . . . 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 1186 . . 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 2436 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
30 eqid 2436 . . . . . . 7  |-  ( 1r
`  S )  =  ( 1r `  S
)
3110, 11, 29, 30unitlinv 15782 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
329, 31sylan 458 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  S ) X )  =  ( 1r `  S ) )
333, 26ressmulr 13582 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3433adantr 452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( .r `  R )  =  ( .r `  S
) )
3534oveqd 6098 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( ( J `
 X ) ( .r `  S ) X ) )
36 eqid 2436 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
373, 36subrg1 15878 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
3837adantr 452 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( 1r `  R )  =  ( 1r `  S
) )
3932, 35, 383eqtr4d 2478 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
4039oveq1d 6096 . . 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 15783 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  (Unit `  R )
)  ->  ( X
( .r `  R
) ( I `  X ) )  =  ( 1r `  R
) )
421, 41sylan 458 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  (Unit `  R ) )  ->  ( X ( .r `  R ) ( I `  X
) )  =  ( 1r `  R ) )
4321, 42syldan 457 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
4443oveq2d 6097 . . 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 2476 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( ( J `
 X ) ( .r `  R ) ( 1r `  R
) ) )
465, 26, 36rnglidm 15687 . . 3  |-  ( ( R  e.  Ring  /\  (
I `  X )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
472, 25, 46syl2anc 643 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) ( I `
 X ) )  =  ( I `  X ) )
485, 26, 36rngridm 15688 . . 3  |-  ( ( R  e.  Ring  /\  ( J `  X )  e.  ( Base `  R
) )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
492, 15, 48syl2anc 643 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
( J `  X
) ( .r `  R ) ( 1r
`  R ) )  =  ( J `  X ) )
5045, 47, 493eqtr3d 2476 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  U )  ->  (
I `  X )  =  ( J `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   .rcmulr 13530   Ringcrg 15660   1rcur 15662  Unitcui 15744   invrcinvr 15776  SubRingcsubrg 15864
This theorem is referenced by:  subrgdv  15885  subrgunit  15886  subrgugrp  15887  issubdrg  15893  gzrngunit  16764
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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