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Theorem divval 9442
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 3762 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
2 eqeq2 2305 . . . . 5  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
32riotabidv 6322 . . . 4  |-  ( z  =  A  ->  ( iota_ x  e.  CC ( y  x.  x )  =  z )  =  ( iota_ x  e.  CC ( y  x.  x
)  =  A ) )
4 oveq1 5881 . . . . . 6  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
54eqeq1d 2304 . . . . 5  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
65riotabidv 6322 . . . 4  |-  ( y  =  B  ->  ( iota_ x  e.  CC ( y  x.  x )  =  A )  =  ( iota_ x  e.  CC ( B  x.  x
)  =  A ) )
7 df-div 9440 . . . 4  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC ( y  x.  x )  =  z ) )
8 riotaex 6324 . . . 4  |-  ( iota_ x  e.  CC ( B  x.  x )  =  A )  e.  _V
93, 6, 7, 8ovmpt2 5999 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ( CC  \  { 0 } ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC ( B  x.  x
)  =  A ) )
101, 9sylan2br 462 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC ( B  x.  x
)  =  A ) )
11103impb 1147 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653  (class class class)co 5874   iota_crio 6313   CCcc 8751   0cc0 8753    x. cmul 8758    / cdiv 9439
This theorem is referenced by:  divmul  9443  divcl  9446  cnflddiv  16420  divcn  18388  rexdiv  23125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-div 9440
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