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Theorem znval 16491
Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.)
Hypotheses
Ref Expression
znval.z  |-  Z  =  (flds  ZZ )
znval.s  |-  S  =  (RSpan `  Z )
znval.u  |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )
znval.y  |-  Y  =  (ℤ/n `  N )
znval.f  |-  F  =  ( ( ZRHom `  U )  |`  W )
znval.w  |-  W  =  if ( N  =  0 ,  ZZ , 
( 0..^ N ) )
znval.l  |-  .<_  =  ( ( F  o.  <_  )  o.  `' F )
Assertion
Ref Expression
znval  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )

Proof of Theorem znval
Dummy variables  f  n  s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 znval.y . 2  |-  Y  =  (ℤ/n `  N )
2 ovex 5885 . . . . 5  |-  (flds  ZZ )  e.  _V
32a1i 10 . . . 4  |-  ( n  =  N  ->  (flds  ZZ )  e.  _V )
4 ovex 5885 . . . . . 6  |-  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  e.  _V
54a1i 10 . . . . 5  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) )  e.  _V )
6 id 19 . . . . . . 7  |-  ( s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) )  ->  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )
7 simpr 447 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
z  =  (flds  ZZ ) )
8 znval.z . . . . . . . . . 10  |-  Z  =  (flds  ZZ )
97, 8syl6eqr 2335 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
z  =  Z )
109fveq2d 5531 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
(RSpan `  z )  =  (RSpan `  Z )
)
11 znval.s . . . . . . . . . . . 12  |-  S  =  (RSpan `  Z )
1210, 11syl6eqr 2335 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
(RSpan `  z )  =  S )
13 simpl 443 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  ->  n  =  N )
1413sneqd 3655 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  ->  { n }  =  { N } )
1512, 14fveq12d 5533 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( (RSpan `  z
) `  { n } )  =  ( S `  { N } ) )
169, 15oveq12d 5878 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( z ~QG  ( (RSpan `  z
) `  { n } ) )  =  ( Z ~QG  ( S `  { N } ) ) )
179, 16oveq12d 5878 . . . . . . . 8  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) )  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) ) )
18 znval.u . . . . . . . 8  |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )
1917, 18syl6eqr 2335 . . . . . . 7  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) )  =  U )
206, 19sylan9eqr 2339 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
s  =  U )
21 fvex 5541 . . . . . . . . . 10  |-  ( ZRHom `  s )  e.  _V
2221resex 4997 . . . . . . . . 9  |-  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ , 
( 0..^ n ) ) )  e.  _V
2322a1i 10 . . . . . . . 8  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  e.  _V )
24 id 19 . . . . . . . . . . . 12  |-  ( f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  ->  f  =  ( ( ZRHom `  s
)  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )
2520fveq2d 5531 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( ZRHom `  s
)  =  ( ZRHom `  U ) )
26 simpll 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  ->  n  =  N )
2726eqeq1d 2293 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( n  =  0  <-> 
N  =  0 ) )
2826oveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( 0..^ n )  =  ( 0..^ N ) )
2927, 28ifbieq2d 3587 . . . . . . . . . . . . . . 15  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  ->  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  if ( N  =  0 ,  ZZ ,  ( 0..^ N ) ) )
30 znval.w . . . . . . . . . . . . . . 15  |-  W  =  if ( N  =  0 ,  ZZ , 
( 0..^ N ) )
3129, 30syl6eqr 2335 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  ->  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  W )
3225, 31reseq12d 4958 . . . . . . . . . . . . 13  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  ( ( ZRHom `  U )  |`  W ) )
33 znval.f . . . . . . . . . . . . 13  |-  F  =  ( ( ZRHom `  U )  |`  W )
3432, 33syl6eqr 2335 . . . . . . . . . . . 12  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  F )
3524, 34sylan9eqr 2339 . . . . . . . . . . 11  |-  ( ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  f  =  F )
3635coeq1d 4847 . . . . . . . . . 10  |-  ( ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  ( f  o.  <_  )  =  ( F  o.  <_  )
)
3735cnveqd 4859 . . . . . . . . . 10  |-  ( ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  `' f  =  `' F )
3836, 37coeq12d 4850 . . . . . . . . 9  |-  ( ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  ( (
f  o.  <_  )  o.  `' f )  =  ( ( F  o.  <_  )  o.  `' F
) )
39 znval.l . . . . . . . . 9  |-  .<_  =  ( ( F  o.  <_  )  o.  `' F )
4038, 39syl6eqr 2335 . . . . . . . 8  |-  ( ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  ( (
f  o.  <_  )  o.  `' f )  = 
.<_  )
4123, 40csbied 3125 . . . . . . 7  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  ->  [_ ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  /  f ]_ (
( f  o.  <_  )  o.  `' f )  =  .<_  )
4241opeq2d 3805 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  ->  <. ( le `  ndx ) ,  [_ ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ , 
( 0..^ n ) ) )  /  f ]_ ( ( f  o. 
<_  )  o.  `' f ) >.  =  <. ( le `  ndx ) ,  .<_  >. )
4320, 42oveq12d 5878 . . . . 5  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( s sSet  <. ( le `  ndx ) , 
[_ ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  /  f ]_ (
( f  o.  <_  )  o.  `' f )
>. )  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
445, 43csbied 3125 . . . 4  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  ->  [_ ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) )  /  s ]_ ( s sSet  <. ( le `  ndx ) , 
[_ ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  /  f ]_ (
( f  o.  <_  )  o.  `' f )
>. )  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
453, 44csbied 3125 . . 3  |-  ( n  =  N  ->  [_ (flds  ZZ )  /  z ]_ [_ (
z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  /  s ]_ (
s sSet  <. ( le `  ndx ) ,  [_ (
( ZRHom `  s
)  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  / 
f ]_ ( ( f  o.  <_  )  o.  `' f ) >.
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
46 df-zn 16460 . . 3  |- ℤ/n =  ( n  e. 
NN0  |->  [_ (flds  ZZ )  /  z ]_ [_ ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) )  /  s ]_ ( s sSet  <. ( le `  ndx ) , 
[_ ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  /  f ]_ (
( f  o.  <_  )  o.  `' f )
>. ) )
47 ovex 5885 . . 3  |-  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
4845, 46, 47fvmpt 5604 . 2  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
491, 48syl5eq 2329 1  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   _Vcvv 2790   [_csb 3083   ifcif 3567   {csn 3642   <.cop 3645   `'ccnv 4690    |` cres 4693    o. ccom 4695   ` cfv 5257  (class class class)co 5860   0cc0 8739    <_ cle 8870   NN0cn0 9967   ZZcz 10026  ..^cfzo 10872   ndxcnx 13147   sSet csts 13148   ↾s cress 13151   lecple 13217    /.s cqus 13410   ~QG cqg 14619  RSpancrsp 15926  ℂfldccnfld 16379   ZRHomczrh 16453  ℤ/nczn 16456
This theorem is referenced by:  znle  16492  znval2  16493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-res 4703  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-zn 16460
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