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Theorem dvrvald 13445
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
dvrvald.b  |-  ( ph  ->  B  =  ( Base `  R ) )
dvrvald.t  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
dvrvald.u  |-  ( ph  ->  U  =  (Unit `  R ) )
dvrvald.i  |-  ( ph  ->  I  =  ( invr `  R ) )
dvrvald.d  |-  ( ph  -> 
./  =  (/r `  R
) )
dvrvald.r  |-  ( ph  ->  R  e.  Ring )
dvrvald.x  |-  ( ph  ->  X  e.  B )
dvrvald.y  |-  ( ph  ->  Y  e.  U )
Assertion
Ref Expression
dvrvald  |-  ( ph  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )

Proof of Theorem dvrvald
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvrvald.b . . 3  |-  ( ph  ->  B  =  ( Base `  R ) )
2 dvrvald.t . . 3  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
3 dvrvald.u . . 3  |-  ( ph  ->  U  =  (Unit `  R ) )
4 dvrvald.i . . 3  |-  ( ph  ->  I  =  ( invr `  R ) )
5 dvrvald.d . . 3  |-  ( ph  -> 
./  =  (/r `  R
) )
6 dvrvald.r . . . 4  |-  ( ph  ->  R  e.  Ring )
7 ringsrg 13360 . . . 4  |-  ( R  e.  Ring  ->  R  e. SRing
)
86, 7syl 14 . . 3  |-  ( ph  ->  R  e. SRing )
91, 2, 3, 4, 5, 8dvrfvald 13444 . 2  |-  ( ph  -> 
./  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) ) )
10 simpl 109 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
11 fveq2 5530 . . . . 5  |-  ( y  =  Y  ->  (
I `  y )  =  ( I `  Y ) )
1211adantl 277 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( I `  y
)  =  ( I `
 Y ) )
1310, 12oveq12d 5909 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  .x.  (
I `  y )
)  =  ( X 
.x.  ( I `  Y ) ) )
1413adantl 277 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x  .x.  (
I `  y )
)  =  ( X 
.x.  ( I `  Y ) ) )
15 dvrvald.x . 2  |-  ( ph  ->  X  e.  B )
16 dvrvald.y . 2  |-  ( ph  ->  Y  e.  U )
172oveqd 5908 . . 3  |-  ( ph  ->  ( X  .x.  (
I `  Y )
)  =  ( X ( .r `  R
) ( I `  Y ) ) )
1815, 1eleqtrd 2268 . . . 4  |-  ( ph  ->  X  e.  ( Base `  R ) )
19 eqidd 2190 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
2016, 3eleqtrd 2268 . . . . . . 7  |-  ( ph  ->  Y  e.  (Unit `  R ) )
21 eqid 2189 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
22 eqid 2189 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2321, 22unitinvcl 13434 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Y  e.  (Unit `  R )
)  ->  ( ( invr `  R ) `  Y )  e.  (Unit `  R ) )
246, 20, 23syl2anc 411 . . . . . 6  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  (Unit `  R )
)
254fveq1d 5532 . . . . . 6  |-  ( ph  ->  ( I `  Y
)  =  ( (
invr `  R ) `  Y ) )
2624, 25, 33eltr4d 2273 . . . . 5  |-  ( ph  ->  ( I `  Y
)  e.  U )
2719, 3, 8, 26unitcld 13419 . . . 4  |-  ( ph  ->  ( I `  Y
)  e.  ( Base `  R ) )
28 eqid 2189 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
29 eqid 2189 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
3028, 29ringcl 13328 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  R
)  /\  ( I `  Y )  e.  (
Base `  R )
)  ->  ( X
( .r `  R
) ( I `  Y ) )  e.  ( Base `  R
) )
316, 18, 27, 30syl3anc 1249 . . 3  |-  ( ph  ->  ( X ( .r
`  R ) ( I `  Y ) )  e.  ( Base `  R ) )
3217, 31eqeltrd 2266 . 2  |-  ( ph  ->  ( X  .x.  (
I `  Y )
)  e.  ( Base `  R ) )
339, 14, 15, 16, 32ovmpod 6019 1  |-  ( ph  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   ` cfv 5231  (class class class)co 5891   Basecbs 12480   .rcmulr 12556  SRingcsrg 13278   Ringcrg 13311  Unitcui 13398   invrcinvr 13431  /rcdvr 13442
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-pre-ltirr 7941  ax-pre-lttrn 7943  ax-pre-ltadd 7945
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-tpos 6264  df-pnf 8012  df-mnf 8013  df-ltxr 8015  df-inn 8938  df-2 8996  df-3 8997  df-ndx 12483  df-slot 12484  df-base 12486  df-sets 12487  df-iress 12488  df-plusg 12568  df-mulr 12569  df-0g 12729  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-minusg 12915  df-cmn 13186  df-abl 13187  df-mgp 13236  df-ur 13275  df-srg 13279  df-ring 13313  df-oppr 13379  df-dvdsr 13400  df-unit 13401  df-invr 13432  df-dvr 13443
This theorem is referenced by:  dvrcl  13446  unitdvcl  13447  dvrid  13448  dvr1  13449  dvrass  13450  dvrcan1  13451  dvrdir  13454  rdivmuldivd  13455  ringinvdv  13456  subrgdv  13546
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