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Theorem znval 16706
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 6006 . . . . 5  |-  (flds  ZZ )  e.  _V
32a1i 10 . . . 4  |-  ( n  =  N  ->  (flds  ZZ )  e.  _V )
4 ovex 6006 . . . . . 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 2416 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
z  =  Z )
109fveq2d 5636 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
(RSpan `  z )  =  (RSpan `  Z )
)
11 znval.s . . . . . . . . . . . 12  |-  S  =  (RSpan `  Z )
1210, 11syl6eqr 2416 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
(RSpan `  z )  =  S )
13 simpl 443 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  ->  n  =  N )
1413sneqd 3742 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  ->  { n }  =  { N } )
1512, 14fveq12d 5638 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( (RSpan `  z
) `  { n } )  =  ( S `  { N } ) )
169, 15oveq12d 5999 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( z ~QG  ( (RSpan `  z
) `  { n } ) )  =  ( Z ~QG  ( S `  { N } ) ) )
179, 16oveq12d 5999 . . . . . . . 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 2416 . . . . . . 7  |-  ( ( n  =  N  /\  z  =  (flds  ZZ ) )  -> 
( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) )  =  U )
206, 19sylan9eqr 2420 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
s  =  U )
21 fvex 5646 . . . . . . . . . 10  |-  ( ZRHom `  s )  e.  _V
2221resex 5098 . . . . . . . . 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 5636 . . . . . . . . . . . . . 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 2374 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( n  =  0  <-> 
N  =  0 ) )
2826oveq2d 5997 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  -> 
( 0..^ n )  =  ( 0..^ N ) )
2927, 28ifbieq2d 3674 . . . . . . . . . . . . . . 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 2416 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =  (flds  ZZ ) )  /\  s  =  ( z  /.s  ( z ~QG  (
(RSpan `  z ) `  { n } ) ) ) )  ->  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  W )
3225, 31reseq12d 5059 . . . . . . . . . . . . 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 2416 . . . . . . . . . . . 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 2420 . . . . . . . . . . 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 4948 . . . . . . . . . 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 4960 . . . . . . . . . 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 4951 . . . . . . . . 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 2416 . . . . . . . 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 3209 . . . . . . 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 3905 . . . . . 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 5999 . . . . 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 3209 . . . 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 3209 . . 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 16675 . . 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 6006 . . 3  |-  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
4845, 46, 47fvmpt 5709 . 2  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
491, 48syl5eq 2410 1  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   [_csb 3167   ifcif 3654   {csn 3729   <.cop 3732   `'ccnv 4791    |` cres 4794    o. ccom 4796   ` cfv 5358  (class class class)co 5981   0cc0 8884    <_ cle 9015   NN0cn0 10114   ZZcz 10175  ..^cfzo 11025   ndxcnx 13353   sSet csts 13354   ↾s cress 13357   lecple 13423    /.s cqus 13618   ~QG cqg 14827  RSpancrsp 16134  ℂfldccnfld 16593   ZRHomczrh 16668  ℤ/nczn 16671
This theorem is referenced by:  znle  16707  znval2  16708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-res 4804  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-zn 16675
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