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Theorem xdivval 27271
Description: Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
Assertion
Ref Expression
xdivval  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem xdivval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4147 . . 3  |-  ( B  e.  ( RR  \  { 0 } )  <-> 
( B  e.  RR  /\  B  =/=  0 ) )
2 simpl 457 . . . . . 6  |-  ( ( y  =  A  /\  x  e.  RR* )  -> 
y  =  A )
32eqeq2d 2476 . . . . 5  |-  ( ( y  =  A  /\  x  e.  RR* )  -> 
( ( z xe x )  =  y  <->  ( z xe x )  =  A ) )
43riotabidva 6255 . . . 4  |-  ( y  =  A  ->  ( iota_ x  e.  RR*  (
z xe x )  =  y )  =  ( iota_ x  e. 
RR*  ( z xe x )  =  A ) )
5 simpl 457 . . . . . . 7  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
z  =  B )
65oveq1d 6292 . . . . . 6  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
( z xe x )  =  ( B xe x ) )
76eqeq1d 2464 . . . . 5  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
( ( z xe x )  =  A  <->  ( B xe x )  =  A ) )
87riotabidva 6255 . . . 4  |-  ( z  =  B  ->  ( iota_ x  e.  RR*  (
z xe x )  =  A )  =  ( iota_ x  e. 
RR*  ( B xe x )  =  A ) )
9 df-xdiv 27270 . . . 4  |- /𝑒  =  ( y  e.  RR* ,  z  e.  ( RR  \  {
0 } )  |->  (
iota_ x  e.  RR*  (
z xe x )  =  y ) )
10 riotaex 6242 . . . 4  |-  ( iota_ x  e.  RR*  ( B xe x )  =  A )  e. 
_V
114, 8, 9, 10ovmpt2 6415 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  ( RR  \  {
0 } ) )  ->  ( A /𝑒  B )  =  ( iota_ x  e. 
RR*  ( B xe x )  =  A ) )
121, 11sylan2br 476 . 2  |-  ( ( A  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
13123impb 1187 1  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657    \ cdif 3468   {csn 4022   iota_crio 6237  (class class class)co 6277   RRcr 9482   0cc0 9483   RR*cxr 9618   xecxmu 11308   /𝑒 cxdiv 27269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-xdiv 27270
This theorem is referenced by:  xdivcld  27275  xdivmul  27277  rexdiv  27278  xdivpnfrp  27285
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