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Theorem divval 9680
Description: Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is nonzero. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
divval  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC ( B  x.  x
)  =  A ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem divval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 3927 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
2 eqeq2 2445 . . . . 5  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
32riotabidv 6551 . . . 4  |-  ( z  =  A  ->  ( iota_ x  e.  CC ( y  x.  x )  =  z )  =  ( iota_ x  e.  CC ( y  x.  x
)  =  A ) )
4 oveq1 6088 . . . . . 6  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
54eqeq1d 2444 . . . . 5  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
65riotabidv 6551 . . . 4  |-  ( y  =  B  ->  ( iota_ x  e.  CC ( y  x.  x )  =  A )  =  ( iota_ x  e.  CC ( B  x.  x
)  =  A ) )
7 df-div 9678 . . . 4  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC ( y  x.  x )  =  z ) )
8 riotaex 6553 . . . 4  |-  ( iota_ x  e.  CC ( B  x.  x )  =  A )  e.  _V
93, 6, 7, 8ovmpt2 6209 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ( CC  \  { 0 } ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC ( B  x.  x
)  =  A ) )
101, 9sylan2br 463 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC ( B  x.  x
)  =  A ) )
11103impb 1149 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC ( B  x.  x
)  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814  (class class class)co 6081   iota_crio 6542   CCcc 8988   0cc0 8990    x. cmul 8995    / cdiv 9677
This theorem is referenced by:  divmul  9681  divcl  9684  cnflddiv  16731  divcn  18898  rexdiv  24172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-div 9678
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