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Theorem dvrfval 15791
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 5730 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvrval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5730 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
6 dvrval.u . . . . . 6  |-  U  =  (Unit `  R )
75, 6syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
8 fveq2 5730 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvrval.t . . . . . . 7  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2488 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
11 eqidd 2439 . . . . . 6  |-  ( r  =  R  ->  x  =  x )
12 fveq2 5730 . . . . . . . 8  |-  ( r  =  R  ->  ( invr `  r )  =  ( invr `  R
) )
13 dvrval.i . . . . . . . 8  |-  I  =  ( invr `  R
)
1412, 13syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( invr `  r )  =  I )
1514fveq1d 5732 . . . . . 6  |-  ( r  =  R  ->  (
( invr `  r ) `  y )  =  ( I `  y ) )
1610, 11, 15oveq123d 6104 . . . . 5  |-  ( r  =  R  ->  (
x ( .r `  r ) ( (
invr `  r ) `  y ) )  =  ( x  .x.  (
I `  y )
) )
174, 7, 16mpt2eq123dv 6138 . . . 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 15790 . . . 4  |- /r  =  (
r  e.  _V  |->  ( x  e.  ( Base `  r ) ,  y  e.  (Unit `  r
)  |->  ( x ( .r `  r ) ( ( invr `  r
) `  y )
) ) )
19 fvex 5744 . . . . . 6  |-  ( Base `  R )  e.  _V
203, 19eqeltri 2508 . . . . 5  |-  B  e. 
_V
21 fvex 5744 . . . . . 6  |-  (Unit `  R )  e.  _V
226, 21eqeltri 2508 . . . . 5  |-  U  e. 
_V
2320, 22mpt2ex 6427 . . . 4  |-  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )  e.  _V
2417, 18, 23fvmpt 5808 . . 3  |-  ( R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
25 fvprc 5724 . . . 4  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  (/) )
26 fvprc 5724 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
273, 26syl5eq 2482 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
28 eqid 2438 . . . . . 6  |-  U  =  U
29 mpt2eq12 6136 . . . . . 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 645 . . . . 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 6429 . . . . 5  |-  ( x  e.  (/) ,  y  e.  U  |->  ( x  .x.  ( I `  y
) ) )  =  (/)
3230, 31syl6eq 2486 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) )  =  (/) )
3325, 32eqtr4d 2473 . . 3  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
3424, 33pm2.61i 159 . 2  |-  (/r `  R
)  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )
351, 34eqtri 2458 1  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   Basecbs 13471   .rcmulr 13532  Unitcui 15746   invrcinvr 15778  /rcdvr 15789
This theorem is referenced by:  dvrval  15792  cnflddiv  16733  dvrcn  18215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-dvr 15790
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