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Theorem 1div0 9441
Description: You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that  (/) is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
Assertion
Ref Expression
1div0  |-  ( 1  /  0 )  =  (/)

Proof of Theorem 1div0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-div 9440 . . 3  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
2 riotaex 6324 . . 3  |-  ( iota_ z  e.  CC ( y  x.  z )  =  x )  e.  _V
31, 2dmmpt2 6210 . 2  |-  dom  /  =  ( CC  X.  ( CC  \  { 0 } ) )
4 eqid 2296 . . 3  |-  0  =  0
5 eldifsni 3763 . . . . 5  |-  ( 0  e.  ( CC  \  { 0 } )  ->  0  =/=  0
)
65adantl 452 . . . 4  |-  ( ( 1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )  ->  0  =/=  0 )
76necon2bi 2505 . . 3  |-  ( 0  =  0  ->  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )
84, 7ax-mp 8 . 2  |-  -.  (
1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )
9 ndmovg 6019 . 2  |-  ( ( dom  /  =  ( CC  X.  ( CC 
\  { 0 } ) )  /\  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )  -> 
( 1  /  0
)  =  (/) )
103, 8, 9mp2an 653 1  |-  ( 1  /  0 )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   (/)c0 3468   {csn 3653    X. cxp 4703   dom cdm 4705  (class class class)co 5874   iota_crio 6313   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-div 9440
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