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Theorem divfnzn 9365
Description: Division restricted to  ZZ  X.  NN is a function. Given excluded middle, it would be easy to prove this for  CC 
X.  ( CC  \  { 0 } ). The key difference is that an element of  NN is apart from zero, whereas being an element of 
CC  \  { 0 } implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
Assertion
Ref Expression
divfnzn  |-  (  /  |`  ( ZZ  X.  NN ) )  Fn  ( ZZ  X.  NN )

Proof of Theorem divfnzn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zcn 9013 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
21ad2antrr 477 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  x  e.  CC )
3 nncn 8688 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
43ad2antlr 478 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  y  e.  CC )
5 simpr 109 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  z  e.  CC )
6 nnap0 8709 . . . . . . 7  |-  ( y  e.  NN  ->  y #  0 )
76ad2antlr 478 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  y #  0 )
82, 4, 5, 7divmulapd 8535 . . . . 5  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  ( ( x  /  y )  =  z  <->  ( y  x.  z )  =  x ) )
98riotabidva 5712 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( x  /  y
)  =  z )  =  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
10 eqcom 2117 . . . . . . 7  |-  ( z  =  ( x  / 
y )  <->  ( x  /  y )  =  z )
1110a1i 9 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( z  =  ( x  /  y )  <-> 
( x  /  y
)  =  z ) )
1211riotabidv 5698 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  z  =  ( x  /  y ) )  =  ( iota_ z  e.  CC  ( x  / 
y )  =  z ) )
13 simpl 108 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  x  e.  CC )
143adantl 273 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  y  e.  CC )
156adantl 273 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  y #  0 )
1613, 14, 15divclapd 8513 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  ( x  /  y
)  e.  CC )
17 reueq 2854 . . . . . . . 8  |-  ( ( x  /  y )  e.  CC  <->  E! z  e.  CC  z  =  ( x  /  y ) )
1816, 17sylib 121 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  E! z  e.  CC  z  =  ( x  /  y ) )
19 riotacl 5710 . . . . . . 7  |-  ( E! z  e.  CC  z  =  ( x  / 
y )  ->  ( iota_ z  e.  CC  z  =  ( x  / 
y ) )  e.  CC )
2018, 19syl 14 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  z  =  ( x  /  y ) )  e.  CC )
211, 20sylan 279 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  z  =  ( x  /  y ) )  e.  CC )
2212, 21eqeltrrd 2193 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( x  /  y
)  =  z )  e.  CC )
239, 22eqeltrrd 2193 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( y  x.  z
)  =  x )  e.  CC )
2423rgen2 2493 . 2  |-  A. x  e.  ZZ  A. y  e.  NN  ( iota_ z  e.  CC  ( y  x.  z )  =  x )  e.  CC
25 df-div 8396 . . . . 5  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
2625reseq1i 4783 . . . 4  |-  (  /  |`  ( ZZ  X.  NN ) )  =  ( ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )  |`  ( ZZ  X.  NN ) )
27 zsscn 9016 . . . . 5  |-  ZZ  C_  CC
28 nncn 8688 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  CC )
29 nnne0 8708 . . . . . . 7  |-  ( x  e.  NN  ->  x  =/=  0 )
30 eldifsn 3618 . . . . . . 7  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
3128, 29, 30sylanbrc 411 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  ( CC  \  {
0 } ) )
3231ssriv 3069 . . . . 5  |-  NN  C_  ( CC  \  { 0 } )
33 resmpo 5835 . . . . 5  |-  ( ( ZZ  C_  CC  /\  NN  C_  ( CC  \  {
0 } ) )  ->  ( ( x  e.  CC ,  y  e.  ( CC  \  { 0 } ) 
|->  ( iota_ z  e.  CC  ( y  x.  z
)  =  x ) )  |`  ( ZZ  X.  NN ) )  =  ( x  e.  ZZ ,  y  e.  NN  |->  ( iota_ z  e.  CC  ( y  x.  z
)  =  x ) ) )
3427, 32, 33mp2an 420 . . . 4  |-  ( ( x  e.  CC , 
y  e.  ( CC 
\  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )  |`  ( ZZ  X.  NN ) )  =  ( x  e.  ZZ ,  y  e.  NN  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
3526, 34eqtri 2136 . . 3  |-  (  /  |`  ( ZZ  X.  NN ) )  =  ( x  e.  ZZ , 
y  e.  NN  |->  (
iota_ z  e.  CC  ( y  x.  z
)  =  x ) )
3635fnmpo 6066 . 2  |-  ( A. x  e.  ZZ  A. y  e.  NN  ( iota_ z  e.  CC  ( y  x.  z )  =  x )  e.  CC  ->  (  /  |`  ( ZZ  X.  NN ) )  Fn  ( ZZ  X.  NN ) )
3724, 36ax-mp 5 1  |-  (  /  |`  ( ZZ  X.  NN ) )  Fn  ( ZZ  X.  NN )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463    =/= wne 2283   A.wral 2391   E!wreu 2393    \ cdif 3036    C_ wss 3039   {csn 3495   class class class wbr 3897    X. cxp 4505    |` cres 4509    Fn wfn 5086   iota_crio 5695  (class class class)co 5740    e. cmpo 5742   CCcc 7582   0cc0 7584    x. cmul 7589   # cap 8306    / cdiv 8395   NNcn 8680   ZZcz 9008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-z 9009
This theorem is referenced by:  elq  9366  qnnen  11850
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