MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvrfval Structured version   Visualization version   Unicode version

Theorem dvrfval 17990
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
dvrval.b  |-  B  =  ( Base `  R
)
dvrval.t  |-  .x.  =  ( .r `  R )
dvrval.u  |-  U  =  (Unit `  R )
dvrval.i  |-  I  =  ( invr `  R
)
dvrval.d  |-  ./  =  (/r
`  R )
Assertion
Ref Expression
dvrfval  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
Distinct variable groups:    x, y, B    x, I, y    x, R, y    x,  .x. , y    x, U, y
Allowed substitution hints:    ./ ( x, y)

Proof of Theorem dvrfval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 dvrval.d . 2  |-  ./  =  (/r
`  R )
2 fveq2 5879 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvrval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2523 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5879 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
6 dvrval.u . . . . . 6  |-  U  =  (Unit `  R )
75, 6syl6eqr 2523 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
8 fveq2 5879 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvrval.t . . . . . . 7  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2523 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
11 eqidd 2472 . . . . . 6  |-  ( r  =  R  ->  x  =  x )
12 fveq2 5879 . . . . . . . 8  |-  ( r  =  R  ->  ( invr `  r )  =  ( invr `  R
) )
13 dvrval.i . . . . . . . 8  |-  I  =  ( invr `  R
)
1412, 13syl6eqr 2523 . . . . . . 7  |-  ( r  =  R  ->  ( invr `  r )  =  I )
1514fveq1d 5881 . . . . . 6  |-  ( r  =  R  ->  (
( invr `  r ) `  y )  =  ( I `  y ) )
1610, 11, 15oveq123d 6329 . . . . 5  |-  ( r  =  R  ->  (
x ( .r `  r ) ( (
invr `  r ) `  y ) )  =  ( x  .x.  (
I `  y )
) )
174, 7, 16mpt2eq123dv 6372 . . . 4  |-  ( r  =  R  ->  (
x  e.  ( Base `  r ) ,  y  e.  (Unit `  r
)  |->  ( x ( .r `  r ) ( ( invr `  r
) `  y )
) )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
18 df-dvr 17989 . . . 4  |- /r  =  (
r  e.  _V  |->  ( x  e.  ( Base `  r ) ,  y  e.  (Unit `  r
)  |->  ( x ( .r `  r ) ( ( invr `  r
) `  y )
) ) )
19 fvex 5889 . . . . . 6  |-  ( Base `  R )  e.  _V
203, 19eqeltri 2545 . . . . 5  |-  B  e. 
_V
21 fvex 5889 . . . . . 6  |-  (Unit `  R )  e.  _V
226, 21eqeltri 2545 . . . . 5  |-  U  e. 
_V
2320, 22mpt2ex 6889 . . . 4  |-  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )  e.  _V
2417, 18, 23fvmpt 5963 . . 3  |-  ( R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
25 fvprc 5873 . . . 4  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  (/) )
26 fvprc 5873 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
273, 26syl5eq 2517 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
28 eqid 2471 . . . . . 6  |-  U  =  U
29 mpt2eq12 6370 . . . . . 6  |-  ( ( B  =  (/)  /\  U  =  U )  ->  (
x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  U  |->  ( x  .x.  ( I `  y
) ) ) )
3027, 28, 29sylancl 675 . . . . 5  |-  ( -.  R  e.  _V  ->  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  U  |->  ( x  .x.  ( I `  y
) ) ) )
31 mpt20 6380 . . . . 5  |-  ( x  e.  (/) ,  y  e.  U  |->  ( x  .x.  ( I `  y
) ) )  =  (/)
3230, 31syl6eq 2521 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) )  =  (/) )
3325, 32eqtr4d 2508 . . 3  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
3424, 33pm2.61i 169 . 2  |-  (/r `  R
)  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )
351, 34eqtri 2493 1  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   Basecbs 15199   .rcmulr 15269  Unitcui 17945   invrcinvr 17977  /rcdvr 17988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-dvr 17989
This theorem is referenced by:  dvrval  17991  cnflddiv  19075  dvrcn  21276
  Copyright terms: Public domain W3C validator