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Theorem divval 9426
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 3749 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
2 eqeq2 2292 . . . . 5  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
32riotabidv 6306 . . . 4  |-  ( z  =  A  ->  ( iota_ x  e.  CC ( y  x.  x )  =  z )  =  ( iota_ x  e.  CC ( y  x.  x
)  =  A ) )
4 oveq1 5865 . . . . . 6  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
54eqeq1d 2291 . . . . 5  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
65riotabidv 6306 . . . 4  |-  ( y  =  B  ->  ( iota_ x  e.  CC ( y  x.  x )  =  A )  =  ( iota_ x  e.  CC ( B  x.  x
)  =  A ) )
7 df-div 9424 . . . 4  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC ( y  x.  x )  =  z ) )
8 riotaex 6308 . . . 4  |-  ( iota_ x  e.  CC ( B  x.  x )  =  A )  e.  _V
93, 6, 7, 8ovmpt2 5983 . . 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640  (class class class)co 5858   iota_crio 6297   CCcc 8735   0cc0 8737    x. cmul 8742    / cdiv 9423
This theorem is referenced by:  divmul  9427  divcl  9430  cnflddiv  16404  divcn  18372  rexdiv  23109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-div 9424
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