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

Theorem dvrvald 13382
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 13297 . . . 4  |-  ( R  e.  Ring  ->  R  e. SRing
)
86, 7syl 14 . . 3  |-  ( ph  ->  R  e. SRing )
91, 2, 3, 4, 5, 8dvrfvald 13381 . 2  |-  ( ph  -> 
./  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) ) )
10 simpl 109 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
11 fveq2 5527 . . . . 5  |-  ( y  =  Y  ->  (
I `  y )  =  ( I `  Y ) )
1211adantl 277 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( I `  y
)  =  ( I `
 Y ) )
1310, 12oveq12d 5906 . . 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 5905 . . 3  |-  ( ph  ->  ( X  .x.  (
I `  Y )
)  =  ( X ( .r `  R
) ( I `  Y ) ) )
1815, 1eleqtrd 2266 . . . 4  |-  ( ph  ->  X  e.  ( Base `  R ) )
19 eqidd 2188 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
2016, 3eleqtrd 2266 . . . . . . 7  |-  ( ph  ->  Y  e.  (Unit `  R ) )
21 eqid 2187 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
22 eqid 2187 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2321, 22unitinvcl 13371 . . . . . . 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 5529 . . . . . 6  |-  ( ph  ->  ( I `  Y
)  =  ( (
invr `  R ) `  Y ) )
2624, 25, 33eltr4d 2271 . . . . 5  |-  ( ph  ->  ( I `  Y
)  e.  U )
2719, 3, 8, 26unitcld 13356 . . . 4  |-  ( ph  ->  ( I `  Y
)  e.  ( Base `  R ) )
28 eqid 2187 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
29 eqid 2187 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
3028, 29ringcl 13265 . . . 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 1248 . . 3  |-  ( ph  ->  ( X ( .r
`  R ) ( I `  Y ) )  e.  ( Base `  R ) )
3217, 31eqeltrd 2264 . 2  |-  ( ph  ->  ( X  .x.  (
I `  Y )
)  e.  ( Base `  R ) )
339, 14, 15, 16, 32ovmpod 6016 1  |-  ( ph  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12476   .rcmulr 12552  SRingcsrg 13215   Ringcrg 13248  Unitcui 13335   invrcinvr 13368  /rcdvr 13379
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-ltadd 7941
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-tpos 6260  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-plusg 12564  df-mulr 12565  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12902  df-minusg 12903  df-cmn 13123  df-abl 13124  df-mgp 13173  df-ur 13212  df-srg 13216  df-ring 13250  df-oppr 13316  df-dvdsr 13337  df-unit 13338  df-invr 13369  df-dvr 13380
This theorem is referenced by:  dvrcl  13383  unitdvcl  13384  dvrid  13385  dvr1  13386  dvrass  13387  dvrcan1  13388  dvrdir  13391  rdivmuldivd  13392  ringinvdv  13393  subrgdv  13458
  Copyright terms: Public domain W3C validator