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Theorem dvrfval 17459
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 5872 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvrval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5872 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
6 dvrval.u . . . . . 6  |-  U  =  (Unit `  R )
75, 6syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
8 fveq2 5872 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvrval.t . . . . . . 7  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2516 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
11 eqidd 2458 . . . . . 6  |-  ( r  =  R  ->  x  =  x )
12 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  ( invr `  r )  =  ( invr `  R
) )
13 dvrval.i . . . . . . . 8  |-  I  =  ( invr `  R
)
1412, 13syl6eqr 2516 . . . . . . 7  |-  ( r  =  R  ->  ( invr `  r )  =  I )
1514fveq1d 5874 . . . . . 6  |-  ( r  =  R  ->  (
( invr `  r ) `  y )  =  ( I `  y ) )
1610, 11, 15oveq123d 6317 . . . . 5  |-  ( r  =  R  ->  (
x ( .r `  r ) ( (
invr `  r ) `  y ) )  =  ( x  .x.  (
I `  y )
) )
174, 7, 16mpt2eq123dv 6358 . . . 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 17458 . . . 4  |- /r  =  (
r  e.  _V  |->  ( x  e.  ( Base `  r ) ,  y  e.  (Unit `  r
)  |->  ( x ( .r `  r ) ( ( invr `  r
) `  y )
) ) )
19 fvex 5882 . . . . . 6  |-  ( Base `  R )  e.  _V
203, 19eqeltri 2541 . . . . 5  |-  B  e. 
_V
21 fvex 5882 . . . . . 6  |-  (Unit `  R )  e.  _V
226, 21eqeltri 2541 . . . . 5  |-  U  e. 
_V
2320, 22mpt2ex 6876 . . . 4  |-  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )  e.  _V
2417, 18, 23fvmpt 5956 . . 3  |-  ( R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
25 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  (/) )
26 fvprc 5866 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
273, 26syl5eq 2510 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
28 eqid 2457 . . . . . 6  |-  U  =  U
29 mpt2eq12 6356 . . . . . 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 662 . . . . 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 6366 . . . . 5  |-  ( x  e.  (/) ,  y  e.  U  |->  ( x  .x.  ( I `  y
) ) )  =  (/)
3230, 31syl6eq 2514 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) )  =  (/) )
3325, 32eqtr4d 2501 . . 3  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
3424, 33pm2.61i 164 . 2  |-  (/r `  R
)  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )
351, 34eqtri 2486 1  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14643   .rcmulr 14712  Unitcui 17414   invrcinvr 17446  /rcdvr 17457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-dvr 17458
This theorem is referenced by:  dvrval  17460  cnflddiv  18574  dvrcn  20811
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