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Theorem xdivval 28053
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 4096 . . 3  |-  ( B  e.  ( RR  \  { 0 } )  <-> 
( B  e.  RR  /\  B  =/=  0 ) )
2 simpl 455 . . . . . 6  |-  ( ( y  =  A  /\  x  e.  RR* )  -> 
y  =  A )
32eqeq2d 2416 . . . . 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 455 . . . . . . 7  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
z  =  B )
65oveq1d 6292 . . . . . 6  |-  ( ( z  =  B  /\  x  e.  RR* )  -> 
( z xe x )  =  ( B xe x ) )
76eqeq1d 2404 . . . . 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 28052 . . . 4  |- /𝑒  =  ( y  e.  RR* ,  z  e.  ( RR  \  {
0 } )  |->  (
iota_ x  e.  RR*  (
z xe x )  =  y ) )
10 riotaex 6243 . . . 4  |-  ( iota_ x  e.  RR*  ( B xe x )  =  A )  e. 
_V
114, 8, 9, 10ovmpt2 6418 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  ( RR  \  {
0 } ) )  ->  ( A /𝑒  B )  =  ( iota_ x  e. 
RR*  ( B xe x )  =  A ) )
121, 11sylan2br 474 . 2  |-  ( ( A  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  -> 
( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
13123impb 1193 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3410   {csn 3971   iota_crio 6238  (class class class)co 6277   RRcr 9520   0cc0 9521   RR*cxr 9656   xecxmu 11369   /𝑒 cxdiv 28051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-xdiv 28052
This theorem is referenced by:  xdivcld  28057  xdivmul  28059  rexdiv  28060  xdivpnfrp  28067
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