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Theorem divfnzn 9075
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 8725 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
21ad2antrr 472 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  x  e.  CC )
3 nncn 8402 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
43ad2antlr 473 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  y  e.  CC )
5 simpr 108 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  z  e.  CC )
6 nnap0 8423 . . . . . . 7  |-  ( y  e.  NN  ->  y #  0 )
76ad2antlr 473 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  y #  0 )
82, 4, 5, 7divmulapd 8252 . . . . 5  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  z  e.  CC )  ->  ( ( x  /  y )  =  z  <->  ( y  x.  z )  =  x ) )
98riotabidva 5606 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( x  /  y
)  =  z )  =  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
10 eqcom 2090 . . . . . . 7  |-  ( z  =  ( x  / 
y )  <->  ( x  /  y )  =  z )
1110a1i 9 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( z  =  ( x  /  y )  <-> 
( x  /  y
)  =  z ) )
1211riotabidv 5592 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  z  =  ( x  /  y ) )  =  ( iota_ z  e.  CC  ( x  / 
y )  =  z ) )
13 simpl 107 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  x  e.  CC )
143adantl 271 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  y  e.  CC )
156adantl 271 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  y #  0 )
1613, 14, 15divclapd 8230 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  ( x  /  y
)  e.  CC )
17 reueq 2812 . . . . . . . 8  |-  ( ( x  /  y )  e.  CC  <->  E! z  e.  CC  z  =  ( x  /  y ) )
1816, 17sylib 120 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  NN )  ->  E! z  e.  CC  z  =  ( x  /  y ) )
19 riotacl 5604 . . . . . . 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 277 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  z  =  ( x  /  y ) )  e.  CC )
2212, 21eqeltrrd 2165 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( x  /  y
)  =  z )  e.  CC )
239, 22eqeltrrd 2165 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( iota_ z  e.  CC  ( y  x.  z
)  =  x )  e.  CC )
2423rgen2 2459 . 2  |-  A. x  e.  ZZ  A. y  e.  NN  ( iota_ z  e.  CC  ( y  x.  z )  =  x )  e.  CC
25 df-div 8114 . . . . 5  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )
2625reseq1i 4697 . . . 4  |-  (  /  |`  ( ZZ  X.  NN ) )  =  ( ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC  ( y  x.  z )  =  x ) )  |`  ( ZZ  X.  NN ) )
27 zsscn 8728 . . . . 5  |-  ZZ  C_  CC
28 nncn 8402 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  CC )
29 nnne0 8422 . . . . . . 7  |-  ( x  e.  NN  ->  x  =/=  0 )
30 eldifsn 3562 . . . . . . 7  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
3128, 29, 30sylanbrc 408 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  ( CC  \  {
0 } ) )
3231ssriv 3027 . . . . 5  |-  NN  C_  ( CC  \  { 0 } )
33 resmpt2 5725 . . . . 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 417 . . . 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 2108 . . 3  |-  (  /  |`  ( ZZ  X.  NN ) )  =  ( x  e.  ZZ , 
y  e.  NN  |->  (
iota_ z  e.  CC  ( y  x.  z
)  =  x ) )
3635fnmpt2 5954 . 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 7 1  |-  (  /  |`  ( ZZ  X.  NN ) )  Fn  ( ZZ  X.  NN )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438    =/= wne 2255   A.wral 2359   E!wreu 2361    \ cdif 2994    C_ wss 2997   {csn 3441   class class class wbr 3837    X. cxp 4426    |` cres 4430    Fn wfn 4997   iota_crio 5589  (class class class)co 5634    |-> cmpt2 5636   CCcc 7327   0cc0 7329    x. cmul 7334   # cap 8034    / cdiv 8113   NNcn 8394   ZZcz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-z 8721
This theorem is referenced by:  elq  9076
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