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Theorem 1div0 9672
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 9671 . . 3  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
2 riotaex 6546 . . 3  |-  ( iota_ z  e.  CC ( y  x.  z )  =  x )  e.  _V
31, 2dmmpt2 6414 . 2  |-  dom  /  =  ( CC  X.  ( CC  \  { 0 } ) )
4 eqid 2436 . . 3  |-  0  =  0
5 eldifsni 3921 . . . . 5  |-  ( 0  e.  ( CC  \  { 0 } )  ->  0  =/=  0
)
65adantl 453 . . . 4  |-  ( ( 1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )  ->  0  =/=  0 )
76necon2bi 2645 . . 3  |-  ( 0  =  0  ->  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )
84, 7ax-mp 8 . 2  |-  -.  (
1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )
9 ndmovg 6223 . 2  |-  ( ( dom  /  =  ( CC  X.  ( CC 
\  { 0 } ) )  /\  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )  -> 
( 1  /  0
)  =  (/) )
103, 8, 9mp2an 654 1  |-  ( 1  /  0 )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3310   (/)c0 3621   {csn 3807    X. cxp 4869   dom cdm 4871  (class class class)co 6074   iota_crio 6535   CCcc 8981   0cc0 8983   1c1 8984    x. cmul 8988    / cdiv 9670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-div 9671
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