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Theorem dvrvald 13341
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 13262 . . . 4  |-  ( R  e.  Ring  ->  R  e. SRing
)
86, 7syl 14 . . 3  |-  ( ph  ->  R  e. SRing )
91, 2, 3, 4, 5, 8dvrfvald 13340 . 2  |-  ( ph  -> 
./  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) ) )
10 simpl 109 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
11 fveq2 5517 . . . . 5  |-  ( y  =  Y  ->  (
I `  y )  =  ( I `  Y ) )
1211adantl 277 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( I `  y
)  =  ( I `
 Y ) )
1310, 12oveq12d 5896 . . 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 5895 . . 3  |-  ( ph  ->  ( X  .x.  (
I `  Y )
)  =  ( X ( .r `  R
) ( I `  Y ) ) )
1815, 1eleqtrd 2256 . . . 4  |-  ( ph  ->  X  e.  ( Base `  R ) )
19 eqidd 2178 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
2016, 3eleqtrd 2256 . . . . . . 7  |-  ( ph  ->  Y  e.  (Unit `  R ) )
21 eqid 2177 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
22 eqid 2177 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2321, 22unitinvcl 13330 . . . . . . 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 5519 . . . . . 6  |-  ( ph  ->  ( I `  Y
)  =  ( (
invr `  R ) `  Y ) )
2624, 25, 33eltr4d 2261 . . . . 5  |-  ( ph  ->  ( I `  Y
)  e.  U )
2719, 3, 8, 26unitcld 13315 . . . 4  |-  ( ph  ->  ( I `  Y
)  e.  ( Base `  R ) )
28 eqid 2177 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
29 eqid 2177 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
3028, 29ringcl 13234 . . . 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 1238 . . 3  |-  ( ph  ->  ( X ( .r
`  R ) ( I `  Y ) )  e.  ( Base `  R ) )
3217, 31eqeltrd 2254 . 2  |-  ( ph  ->  ( X  .x.  (
I `  Y )
)  e.  ( Base `  R ) )
339, 14, 15, 16, 32ovmpod 6005 1  |-  ( ph  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   ` cfv 5218  (class class class)co 5878   Basecbs 12465   .rcmulr 12540  SRingcsrg 13184   Ringcrg 13217  Unitcui 13294   invrcinvr 13327  /rcdvr 13338
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-tpos 6249  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473  df-plusg 12552  df-mulr 12553  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12826  df-grp 12888  df-minusg 12889  df-cmn 13105  df-abl 13106  df-mgp 13147  df-ur 13181  df-srg 13185  df-ring 13219  df-oppr 13278  df-dvdsr 13296  df-unit 13297  df-invr 13328  df-dvr 13339
This theorem is referenced by:  dvrcl  13342  unitdvcl  13343  dvrid  13344  dvr1  13345  dvrass  13346  dvrcan1  13347  dvrdir  13350  rdivmuldivd  13351  ringinvdv  13352  subrgdv  13397
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