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Theorem nZdef 25180
Description: Two ways to define  n ZZ. In the first way I multiply the set  { N } by the set  ZZ ( I think this is this sort of multiplication that is at the origin of the denotation  n ZZ). In the second way I multiply the integer  N by an element of  ZZ. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
nZdef  |-  ( N  e.  ZZ  ->  ( { N }  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) ) ZZ )  =  { x  |  E. y  e.  ZZ  x  =  ( N  x.  y ) } )
Distinct variable group:    x, N, y

Proof of Theorem nZdef
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 snelpwi 4220 . . . 4  |-  ( N  e.  ZZ  ->  { N }  e.  ~P ZZ )
2 ax-mulf 8817 . . . . . . 7  |-  x.  :
( CC  X.  CC )
--> CC
3 zsscn 10032 . . . . . . . 8  |-  ZZ  C_  CC
4 xpss12 4792 . . . . . . . 8  |-  ( ( ZZ  C_  CC  /\  ZZ  C_  CC )  ->  ( ZZ  X.  ZZ )  C_  ( CC  X.  CC ) )
53, 3, 4mp2an 653 . . . . . . 7  |-  ( ZZ 
X.  ZZ )  C_  ( CC  X.  CC )
6 fssres 5408 . . . . . . 7  |-  ( (  x.  : ( CC 
X.  CC ) --> CC 
/\  ( ZZ  X.  ZZ )  C_  ( CC 
X.  CC ) )  ->  (  x.  |`  ( ZZ  X.  ZZ ) ) : ( ZZ  X.  ZZ ) --> CC )
72, 5, 6mp2an 653 . . . . . 6  |-  (  x.  |`  ( ZZ  X.  ZZ ) ) : ( ZZ  X.  ZZ ) --> CC
8 zex 10033 . . . . . . 7  |-  ZZ  e.  _V
98, 8xpex 4801 . . . . . 6  |-  ( ZZ 
X.  ZZ )  e. 
_V
10 cnex 8818 . . . . . 6  |-  CC  e.  _V
11 fex2 5401 . . . . . 6  |-  ( ( (  x.  |`  ( ZZ  X.  ZZ ) ) : ( ZZ  X.  ZZ ) --> CC  /\  ( ZZ  X.  ZZ )  e. 
_V  /\  CC  e.  _V )  ->  (  x.  |`  ( ZZ  X.  ZZ ) )  e.  _V )
127, 9, 10, 11mp3an 1277 . . . . 5  |-  (  x.  |`  ( ZZ  X.  ZZ ) )  e.  _V
138pwid 3638 . . . . 5  |-  ZZ  e.  ~P ZZ
147fdmi 5394 . . . . . . . 8  |-  dom  (  x.  |`  ( ZZ  X.  ZZ ) )  =  ( ZZ  X.  ZZ )
1514dmeqi 4880 . . . . . . 7  |-  dom  dom  (  x.  |`  ( ZZ 
X.  ZZ ) )  =  dom  ( ZZ 
X.  ZZ )
16 dmxpid 4898 . . . . . . 7  |-  dom  ( ZZ  X.  ZZ )  =  ZZ
1715, 16eqtr2i 2304 . . . . . 6  |-  ZZ  =  dom  dom  (  x.  |`  ( ZZ  X.  ZZ ) )
18 eqid 2283 . . . . . 6  |-  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) )  =  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) )
1917, 18iscst3 25176 . . . . 5  |-  ( ( (  x.  |`  ( ZZ  X.  ZZ ) )  e.  _V  /\  { N }  e.  ~P ZZ  /\  ZZ  e.  ~P ZZ )  ->  ( x  e.  ( { N }  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) ) ZZ )  <->  E. z  e.  { N } E. y  e.  ZZ  x  =  ( z (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
2012, 13, 19mp3an13 1268 . . . 4  |-  ( { N }  e.  ~P ZZ  ->  ( x  e.  ( { N } 
( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) ) ZZ )  <->  E. z  e.  { N } E. y  e.  ZZ  x  =  ( z (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
211, 20syl 15 . . 3  |-  ( N  e.  ZZ  ->  (
x  e.  ( { N }  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) ) ZZ )  <->  E. z  e.  { N } E. y  e.  ZZ  x  =  ( z (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
22 oveq1 5865 . . . . . 6  |-  ( z  =  N  ->  (
z (  x.  |`  ( ZZ  X.  ZZ ) ) y )  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y ) )
2322eqeq2d 2294 . . . . 5  |-  ( z  =  N  ->  (
x  =  ( z (  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  x  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
2423rexbidv 2564 . . . 4  |-  ( z  =  N  ->  ( E. y  e.  ZZ  x  =  ( z
(  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  E. y  e.  ZZ  x  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
2524rexsng 3673 . . 3  |-  ( N  e.  ZZ  ->  ( E. z  e.  { N } E. y  e.  ZZ  x  =  ( z
(  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  E. y  e.  ZZ  x  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
26 ovres 5987 . . . . 5  |-  ( ( N  e.  ZZ  /\  y  e.  ZZ )  ->  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y )  =  ( N  x.  y ) )
2726eqeq2d 2294 . . . 4  |-  ( ( N  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  x  =  ( N  x.  y
) ) )
2827rexbidva 2560 . . 3  |-  ( N  e.  ZZ  ->  ( E. y  e.  ZZ  x  =  ( N
(  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  E. y  e.  ZZ  x  =  ( N  x.  y ) ) )
2921, 25, 283bitrd 270 . 2  |-  ( N  e.  ZZ  ->  (
x  e.  ( { N }  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) ) ZZ )  <->  E. y  e.  ZZ  x  =  ( N  x.  y ) ) )
3029abbi2dv 2398 1  |-  ( N  e.  ZZ  ->  ( { N }  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) ) ZZ )  =  { x  |  E. y  e.  ZZ  x  =  ( N  x.  y ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   {csn 3640    X. cxp 4687   dom cdm 4689    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   ZZcz 10024   csetccst 25172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-neg 9040  df-z 10025  df-cst 25173
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