ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subrgdv Unicode version

Theorem subrgdv 13297
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 2177 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
3 subrgdv.3 . . . . . 6  |-  U  =  (Unit `  S )
4 eqid 2177 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
51, 2, 3, 4subrginv 13296 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  =  ( ( invr `  S
) `  Y )
)
653adant2 1016 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( ( invr `  R ) `  Y )  =  ( ( invr `  S
) `  Y )
)
76oveq2d 5888 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) )  =  ( X ( .r `  R ) ( (
invr `  S ) `  Y ) ) )
8 subrgrcl 13285 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
9 eqid 2177 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
101, 9ressmulrg 12595 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
118, 10mpdan 421 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
12113ad2ant1 1018 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( .r `  R )  =  ( .r `  S ) )
1312oveqd 5889 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  S ) `  Y
) )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
147, 13eqtrd 2210 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
15 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( Base `  R )  =  (
Base `  R )
)
16 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( .r `  R )  =  ( .r `  R ) )
17 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  (Unit `  R
)  =  (Unit `  R ) )
18 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( invr `  R )  =  (
invr `  R )
)
19 subrgdv.2 . . . 4  |-  ./  =  (/r
`  R )
2019a1i 9 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ./  =  (/r `  R ) )
2183ad2ant1 1018 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  R  e.  Ring )
22 eqid 2177 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2322subrgss 13281 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
24233ad2ant1 1018 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  A  C_  ( Base `  R ) )
25 simp2 998 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  A )
2624, 25sseldd 3156 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  ( Base `  R )
)
27 eqid 2177 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
281, 27, 3subrguss 13295 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
29283ad2ant1 1018 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  U  C_  (Unit `  R ) )
30 simp3 999 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  Y  e.  U )
3129, 30sseldd 3156 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  Y  e.  (Unit `  R ) )
3215, 16, 17, 18, 20, 21, 26, 31dvrvald 13234 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X ( .r `  R ) ( (
invr `  R ) `  Y ) ) )
33 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( Base `  S )  =  (
Base `  S )
)
34 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( .r `  S )  =  ( .r `  S ) )
353a1i 9 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  U  =  (Unit `  S ) )
36 eqidd 2178 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( invr `  S )  =  (
invr `  S )
)
37 subrgdv.4 . . . 4  |-  E  =  (/r `  S )
3837a1i 9 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  E  =  (/r
`  S ) )
391subrgring 13283 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
40393ad2ant1 1018 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  S  e.  Ring )
411subrgbas 13289 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
42413ad2ant1 1018 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  A  =  ( Base `  S )
)
4325, 42eleqtrd 2256 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  ( Base `  S )
)
4433, 34, 35, 36, 38, 40, 43, 30dvrvald 13234 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X E Y )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
4514, 32, 443eqtr4d 2220 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X E Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   .rcmulr 12529   Ringcrg 13110  Unitcui 13187   invrcinvr 13220  /rcdvr 13231  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-1st 6138  df-2nd 6139  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-dvr 13232  df-subrg 13278
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator