MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divval Structured version   Unicode version

Theorem divval 10210
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 4152 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
2 eqeq2 2482 . . . . 5  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
32riotabidv 6248 . . . 4  |-  ( z  =  A  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  z )  =  ( iota_ x  e.  CC  ( y  x.  x
)  =  A ) )
4 oveq1 6292 . . . . . 6  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
54eqeq1d 2469 . . . . 5  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
65riotabidv 6248 . . . 4  |-  ( y  =  B  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
7 df-div 10208 . . . 4  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC  ( y  x.  x )  =  z ) )
8 riotaex 6250 . . . 4  |-  ( iota_ x  e.  CC  ( B  x.  x )  =  A )  e.  _V
93, 6, 7, 8ovmpt2 6423 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ( CC  \  { 0 } ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
101, 9sylan2br 476 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
11103impb 1192 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   iota_crio 6245  (class class class)co 6285   CCcc 9491   0cc0 9493    x. cmul 9498    / cdiv 10207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-div 10208
This theorem is referenced by:  divmul  10211  divcl  10214  cnflddiv  18259  divcn  21199  rexdiv  27387
  Copyright terms: Public domain W3C validator