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Theorem znval 17966
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.) (Revised by AV, 13-Jun-2019.)
Hypotheses
Ref Expression
znval.s  |-  S  =  (RSpan ` ring )
znval.u  |-  U  =  (ring 
/.s  (ring ~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 zringrng 17886 . . . . 5  |-ring  e.  Ring
32a1i 11 . . . 4  |-  ( n  =  N  ->ring  e.  Ring )
4 ovex 6116 . . . . . 6  |-  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  e.  _V
54a1i 11 . . . . 5  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  e.  _V )
6 id 22 . . . . . . 7  |-  ( s  =  ( z  /.s  (
z ~QG 
( (RSpan `  z
) `  { n } ) ) )  ->  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )
7 simpr 461 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =ring )  ->  z  =ring )
87fveq2d 5695 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  (RSpan ` ring ) )
9 znval.s . . . . . . . . . . . 12  |-  S  =  (RSpan ` ring )
108, 9syl6eqr 2493 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  S )
11 simpl 457 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  n  =  N )
1211sneqd 3889 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  { n }  =  { N } )
1310, 12fveq12d 5697 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =ring )  ->  ( (RSpan `  z ) `  {
n } )  =  ( S `  { N } ) )
147, 13oveq12d 6109 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z ~QG  ( (RSpan `  z ) `  { n } ) )  =  (ring ~QG  ( S `  { N } ) ) )
157, 14oveq12d 6109 . . . . . . . 8  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  =  (ring 
/.s  (ring ~QG  ( S `  { N } ) ) ) )
16 znval.u . . . . . . . 8  |-  U  =  (ring 
/.s  (ring ~QG  ( S `  { N } ) ) )
1715, 16syl6eqr 2493 . . . . . . 7  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  =  U )
186, 17sylan9eqr 2497 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  s  =  U )
19 fvex 5701 . . . . . . . . . 10  |-  ( ZRHom `  s )  e.  _V
2019resex 5150 . . . . . . . . 9  |-  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ , 
( 0..^ n ) ) )  e.  _V
2120a1i 11 . . . . . . . 8  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  e.  _V )
22 id 22 . . . . . . . . . . . 12  |-  ( f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  ->  f  =  ( ( ZRHom `  s
)  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )
2318fveq2d 5695 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ZRHom `  s )  =  ( ZRHom `  U )
)
24 simpll 753 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  n  =  N )
2524eqeq1d 2451 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( n  =  0  <->  N  = 
0 ) )
2624oveq2d 6107 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( 0..^ n )  =  ( 0..^ N ) )
2725, 26ifbieq2d 3814 . . . . . . . . . . . . . . 15  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  if (
n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  if ( N  =  0 ,  ZZ , 
( 0..^ N ) ) )
28 znval.w . . . . . . . . . . . . . . 15  |-  W  =  if ( N  =  0 ,  ZZ , 
( 0..^ N ) )
2927, 28syl6eqr 2493 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  if (
n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  W )
3023, 29reseq12d 5111 . . . . . . . . . . . . 13  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  ( ( ZRHom `  U )  |`  W ) )
31 znval.f . . . . . . . . . . . . 13  |-  F  =  ( ( ZRHom `  U )  |`  W )
3230, 31syl6eqr 2493 . . . . . . . . . . . 12  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  F )
3322, 32sylan9eqr 2497 . . . . . . . . . . 11  |-  ( ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  f  =  F )
3433coeq1d 5001 . . . . . . . . . 10  |-  ( ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  ( f  o.  <_  )  =  ( F  o.  <_  )
)
3533cnveqd 5015 . . . . . . . . . 10  |-  ( ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  `' f  =  `' F )
3634, 35coeq12d 5004 . . . . . . . . 9  |-  ( ( ( ( n  =  N  /\  z  =ring )  /\  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
) )
37 znval.l . . . . . . . . 9  |-  .<_  =  ( ( F  o.  <_  )  o.  `' F )
3836, 37syl6eqr 2493 . . . . . . . 8  |-  ( ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  /\  f  =  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) ) )  ->  ( (
f  o.  <_  )  o.  `' f )  = 
.<_  )
3921, 38csbied 3314 . . . . . . 7  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  [_ ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ , 
( 0..^ n ) ) )  /  f ]_ ( ( f  o. 
<_  )  o.  `' f )  =  .<_  )
4039opeq2d 4066 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =ring )  /\  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 ) ,  .<_  >.
)
4118, 40oveq12d 6109 . . . . 5  |-  ( ( ( n  =  N  /\  z  =ring )  /\  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 ) ,  .<_  >. ) )
425, 41csbied 3314 . . . 4  |-  ( ( n  =  N  /\  z  =ring )  ->  [_ (
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 ) ,  .<_  >. )
)
433, 42csbied 3314 . . 3  |-  ( n  =  N  ->  [_ring  /  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 ) ,  .<_  >. )
)
44 df-zn 17938 . . 3  |- ℤ/n =  ( n  e. 
NN0  |->  [_ring  /  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 ) >.
) )
45 ovex 6116 . . 3  |-  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
4643, 44, 45fvmpt 5774 . 2  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
471, 46syl5eq 2487 1  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   [_csb 3288   ifcif 3791   {csn 3877   <.cop 3883   `'ccnv 4839    |` cres 4842    o. ccom 4844   ` cfv 5418  (class class class)co 6091   0cc0 9282    <_ cle 9419   NN0cn0 10579   ZZcz 10646  ..^cfzo 11548   ndxcnx 14171   sSet csts 14172   lecple 14245    /.s cqus 14443   ~QG cqg 15677   Ringcrg 16645  RSpancrsp 17252  ℤringzring 17883   ZRHomczrh 17931  ℤ/nczn 17934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-subg 15678  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-subrg 16863  df-cnfld 17819  df-zring 17884  df-zn 17938
This theorem is referenced by:  znle  17967  znval2  17970
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