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Theorem divfnzn 9369
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 9017 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
21ad2antrr 479 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  x  e.  CC )
3 nncn 8692 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
43ad2antlr 480 . . . . . 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 8713 . . . . . . 7  |-  ( y  e.  NN  ->  y #  0 )
76ad2antlr 480 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  y #  0 )
82, 4, 5, 7divmulapd 8539 . . . . 5  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  ( ( x  /  y )  =  z  <->  ( y  x.  z )  =  x ) )
98riotabidva 5714 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( x  /  y
)  =  z )  =  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
10 eqcom 2119 . . . . . . 7  |-  ( z  =  ( x  / 
y )  <->  ( x  /  y )  =  z )
1110a1i 9 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( z  =  ( x  /  y )  <-> 
( x  /  y
)  =  z ) )
1211riotabidv 5700 . . . . 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 275 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  y  e.  CC )
156adantl 275 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  y #  0 )
1613, 14, 15divclapd 8517 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  ( x  /  y
)  e.  CC )
17 reueq 2856 . . . . . . . 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 5712 . . . . . . 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 281 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  z  =  ( x  /  y ) )  e.  CC )
2212, 21eqeltrrd 2195 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( x  /  y
)  =  z )  e.  CC )
239, 22eqeltrrd 2195 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( y  x.  z
)  =  x )  e.  CC )
2423rgen2 2495 . 2  |-  A. x  e.  ZZ  A. y  e.  NN  ( iota_ z  e.  CC  ( y  x.  z )  =  x )  e.  CC
25 df-div 8400 . . . . 5  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
2625reseq1i 4785 . . . 4  |-  (  /  |`  ( ZZ  X.  NN ) )  =  ( ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )  |`  ( ZZ  X.  NN ) )
27 zsscn 9020 . . . . 5  |-  ZZ  C_  CC
28 nncn 8692 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  CC )
29 nnne0 8712 . . . . . . 7  |-  ( x  e.  NN  ->  x  =/=  0 )
30 eldifsn 3620 . . . . . . 7  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
3128, 29, 30sylanbrc 413 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  ( CC  \  {
0 } ) )
3231ssriv 3071 . . . . 5  |-  NN  C_  ( CC  \  { 0 } )
33 resmpo 5837 . . . . 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 422 . . . 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 2138 . . 3  |-  (  /  |`  ( ZZ  X.  NN ) )  =  ( x  e.  ZZ , 
y  e.  NN  |->  (
iota_ z  e.  CC  ( y  x.  z
)  =  x ) )
3635fnmpo 6068 . 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 1316    e. wcel 1465    =/= wne 2285   A.wral 2393   E!wreu 2395    \ cdif 3038    C_ wss 3041   {csn 3497   class class class wbr 3899    X. cxp 4507    |` cres 4511    Fn wfn 5088   iota_crio 5697  (class class class)co 5742    e. cmpo 5744   CCcc 7586   0cc0 7588    x. cmul 7593   # cap 8310    / cdiv 8399   NNcn 8684   ZZcz 9012
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-z 9013
This theorem is referenced by:  elq  9370  qnnen  11855
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