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Theorem subrgdv 17222
Description: A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgdv.1  |-  S  =  ( Rs  A )
subrgdv.2  |-  ./  =  (/r
`  R )
subrgdv.3  |-  U  =  (Unit `  S )
subrgdv.4  |-  E  =  (/r `  S )
Assertion
Ref Expression
subrgdv  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X E Y ) )

Proof of Theorem subrgdv
StepHypRef Expression
1 subrgdv.1 . . . . . 6  |-  S  =  ( Rs  A )
2 eqid 2460 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
3 subrgdv.3 . . . . . 6  |-  U  =  (Unit `  S )
4 eqid 2460 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
51, 2, 3, 4subrginv 17221 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  =  ( ( invr `  S
) `  Y )
)
653adant2 1010 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( ( invr `  R ) `  Y )  =  ( ( invr `  S
) `  Y )
)
76oveq2d 6291 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) )  =  ( X ( .r `  R ) ( (
invr `  S ) `  Y ) ) )
8 eqid 2460 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
91, 8ressmulr 14597 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1093ad2ant1 1012 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( .r `  R )  =  ( .r `  S ) )
1110oveqd 6292 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  S ) `  Y
) )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
127, 11eqtrd 2501 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
13 eqid 2460 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
1413subrgss 17206 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
15143ad2ant1 1012 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  A  C_  ( Base `  R ) )
16 simp2 992 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  A )
1715, 16sseldd 3498 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  ( Base `  R )
)
18 eqid 2460 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
191, 18, 3subrguss 17220 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
20193ad2ant1 1012 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  U  C_  (Unit `  R ) )
21 simp3 993 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  Y  e.  U )
2220, 21sseldd 3498 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  Y  e.  (Unit `  R ) )
23 subrgdv.2 . . . 4  |-  ./  =  (/r
`  R )
2413, 8, 18, 2, 23dvrval 17111 . . 3  |-  ( ( X  e.  ( Base `  R )  /\  Y  e.  (Unit `  R )
)  ->  ( X  ./  Y )  =  ( X ( .r `  R ) ( (
invr `  R ) `  Y ) ) )
2517, 22, 24syl2anc 661 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X ( .r `  R ) ( (
invr `  R ) `  Y ) ) )
261subrgbas 17214 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
27263ad2ant1 1012 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  A  =  ( Base `  S )
)
2816, 27eleqtrd 2550 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  ( Base `  S )
)
29 eqid 2460 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
30 eqid 2460 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
31 subrgdv.4 . . . 4  |-  E  =  (/r `  S )
3229, 30, 3, 4, 31dvrval 17111 . . 3  |-  ( ( X  e.  ( Base `  S )  /\  Y  e.  U )  ->  ( X E Y )  =  ( X ( .r
`  S ) ( ( invr `  S
) `  Y )
) )
3328, 21, 32syl2anc 661 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X E Y )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
3412, 25, 333eqtr4d 2511 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X E Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    C_ wss 3469   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480   .rcmulr 14545  Unitcui 17065   invrcinvr 17097  /rcdvr 17108  SubRingcsubrg 17201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-subg 15986  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-subrg 17203
This theorem is referenced by:  qsssubdrg  18238  redvr  18413  cvsdiv  21337  qrngdiv  23530
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