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Theorem divval 10271
Description: Value of division: if  A and  B are complex numbers with  B nonzero, then  ( A  /  B
) is the (unique) complex number such that  ( B  x.  x )  =  A. (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 4128 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
2 eqeq2 2444 . . . . 5  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
32riotabidv 6269 . . . 4  |-  ( z  =  A  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  z )  =  ( iota_ x  e.  CC  ( y  x.  x
)  =  A ) )
4 oveq1 6312 . . . . . 6  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
54eqeq1d 2431 . . . . 5  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
65riotabidv 6269 . . . 4  |-  ( y  =  B  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
7 df-div 10269 . . . 4  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC  ( y  x.  x )  =  z ) )
8 riotaex 6271 . . . 4  |-  ( iota_ x  e.  CC  ( B  x.  x )  =  A )  e.  _V
93, 6, 7, 8ovmpt2 6446 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ( CC  \  { 0 } ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
101, 9sylan2br 478 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
11103impb 1201 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439   {csn 4002   iota_crio 6266  (class class class)co 6305   CCcc 9536   0cc0 9538    x. cmul 9543    / cdiv 10268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-div 10269
This theorem is referenced by:  divmul  10272  divcl  10275  cnflddiv  18933  divcn  21796  rexdiv  28233
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