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Theorem dvrfval 15466
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 5525 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvrval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5525 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
6 dvrval.u . . . . . 6  |-  U  =  (Unit `  R )
75, 6syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
8 fveq2 5525 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvrval.t . . . . . . 7  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2333 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
11 eqidd 2284 . . . . . 6  |-  ( r  =  R  ->  x  =  x )
12 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  ( invr `  r )  =  ( invr `  R
) )
13 dvrval.i . . . . . . . 8  |-  I  =  ( invr `  R
)
1412, 13syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  ( invr `  r )  =  I )
1514fveq1d 5527 . . . . . 6  |-  ( r  =  R  ->  (
( invr `  r ) `  y )  =  ( I `  y ) )
1610, 11, 15oveq123d 5879 . . . . 5  |-  ( r  =  R  ->  (
x ( .r `  r ) ( (
invr `  r ) `  y ) )  =  ( x  .x.  (
I `  y )
) )
174, 7, 16mpt2eq123dv 5910 . . . 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 15465 . . . 4  |- /r  =  (
r  e.  _V  |->  ( x  e.  ( Base `  r ) ,  y  e.  (Unit `  r
)  |->  ( x ( .r `  r ) ( ( invr `  r
) `  y )
) ) )
19 fvex 5539 . . . . . 6  |-  ( Base `  R )  e.  _V
203, 19eqeltri 2353 . . . . 5  |-  B  e. 
_V
21 fvex 5539 . . . . . 6  |-  (Unit `  R )  e.  _V
226, 21eqeltri 2353 . . . . 5  |-  U  e. 
_V
2320, 22mpt2ex 6198 . . . 4  |-  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )  e.  _V
2417, 18, 23fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
25 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  (/) )
26 fvprc 5519 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
273, 26syl5eq 2327 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
28 eqid 2283 . . . . . 6  |-  U  =  U
29 mpt2eq12 5908 . . . . . 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 643 . . . . 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 6199 . . . . 5  |-  ( x  e.  (/) ,  y  e.  U  |->  ( x  .x.  ( I `  y
) ) )  =  (/)
3230, 31syl6eq 2331 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) )  =  (/) )
3325, 32eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (/r `  R )  =  ( x  e.  B , 
y  e.  U  |->  ( x  .x.  ( I `
 y ) ) ) )
3424, 33pm2.61i 156 . 2  |-  (/r `  R
)  =  ( x  e.  B ,  y  e.  U  |->  ( x 
.x.  ( I `  y ) ) )
351, 34eqtri 2303 1  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   .rcmulr 13209  Unitcui 15421   invrcinvr 15453  /rcdvr 15464
This theorem is referenced by:  dvrval  15467  cnflddiv  16404  dvrcn  17866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-dvr 15465
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