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Theorem xdivval 26093
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 3999 . . 3  |-  ( B  e.  ( RR  \  { 0 } )  <-> 
( B  e.  RR  /\  B  =/=  0 ) )
2 simpl 457 . . . . . 6  |-  ( ( y  =  A  /\  x  e.  RR* )  -> 
y  =  A )
32eqeq2d 2453 . . . . 5  |-  ( ( y  =  A  /\  x  e.  RR* )  -> 
( ( z xe x )  =  y  <->  ( z xe x )  =  A ) )
43riotabidva 6068 . . . 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 6105 . . . . . 6  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
( z xe x )  =  ( B xe x ) )
76eqeq1d 2450 . . . . 5  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
( ( z xe x )  =  A  <->  ( B xe x )  =  A ) )
87riotabidva 6068 . . . 4  |-  ( z  =  B  ->  ( iota_ x  e.  RR*  (
z xe x )  =  A )  =  ( iota_ x  e. 
RR*  ( B xe x )  =  A ) )
9 df-xdiv 26092 . . . 4  |- /𝑒  =  ( y  e.  RR* ,  z  e.  ( RR  \  {
0 } )  |->  (
iota_ x  e.  RR*  (
z xe x )  =  y ) )
10 riotaex 6055 . . . 4  |-  ( iota_ x  e.  RR*  ( B xe x )  =  A )  e. 
_V
114, 8, 9, 10ovmpt2 6225 . . 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 1183 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 965    = wceq 1369    e. wcel 1756    =/= wne 2605    \ cdif 3324   {csn 3876   iota_crio 6050  (class class class)co 6090   RRcr 9280   0cc0 9281   RR*cxr 9416   xecxmu 11087   /𝑒 cxdiv 26091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-xdiv 26092
This theorem is referenced by:  xdivcld  26097  xdivmul  26099  rexdiv  26100  xdivpnfrp  26107
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