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Theorem nZdef 25283
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 4236 . . . 4  |-  ( N  e.  ZZ  ->  { N }  e.  ~P ZZ )
2 ax-mulf 8833 . . . . . . 7  |-  x.  :
( CC  X.  CC )
--> CC
3 zsscn 10048 . . . . . . . 8  |-  ZZ  C_  CC
4 xpss12 4808 . . . . . . . 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 5424 . . . . . . 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 10049 . . . . . . 7  |-  ZZ  e.  _V
98, 8xpex 4817 . . . . . 6  |-  ( ZZ 
X.  ZZ )  e. 
_V
10 cnex 8834 . . . . . 6  |-  CC  e.  _V
11 fex2 5417 . . . . . 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 3651 . . . . 5  |-  ZZ  e.  ~P ZZ
147fdmi 5410 . . . . . . . 8  |-  dom  (  x.  |`  ( ZZ  X.  ZZ ) )  =  ( ZZ  X.  ZZ )
1514dmeqi 4896 . . . . . . 7  |-  dom  dom  (  x.  |`  ( ZZ 
X.  ZZ ) )  =  dom  ( ZZ 
X.  ZZ )
16 dmxpid 4914 . . . . . . 7  |-  dom  ( ZZ  X.  ZZ )  =  ZZ
1715, 16eqtr2i 2317 . . . . . 6  |-  ZZ  =  dom  dom  (  x.  |`  ( ZZ  X.  ZZ ) )
18 eqid 2296 . . . . . 6  |-  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) )  =  ( cset `  (  x.  |`  ( ZZ  X.  ZZ ) ) )
1917, 18iscst3 25279 . . . . 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 5881 . . . . . 6  |-  ( z  =  N  ->  (
z (  x.  |`  ( ZZ  X.  ZZ ) ) y )  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y ) )
2322eqeq2d 2307 . . . . 5  |-  ( z  =  N  ->  (
x  =  ( z (  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  x  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y ) ) )
2423rexbidv 2577 . . . 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 3686 . . 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 6003 . . . . 5  |-  ( ( N  e.  ZZ  /\  y  e.  ZZ )  ->  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y )  =  ( N  x.  y ) )
2726eqeq2d 2307 . . . 4  |-  ( ( N  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  =  ( N (  x.  |`  ( ZZ  X.  ZZ ) ) y )  <->  x  =  ( N  x.  y
) ) )
2827rexbidva 2573 . . 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 2411 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {csn 3653    X. cxp 4703   dom cdm 4705    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   ZZcz 10040   csetccst 25275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-neg 9056  df-z 10041  df-cst 25276
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