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Theorem znval 18332
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 18252 . . . . 5  |-ring  e.  Ring
32a1i 11 . . . 4  |-  ( n  =  N  ->ring  e.  Ring )
4 ovex 6300 . . . . . 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 5861 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  (RSpan ` ring ) )
9 znval.s . . . . . . . . . . . 12  |-  S  =  (RSpan ` ring )
108, 9syl6eqr 2519 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  S )
11 simpl 457 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  n  =  N )
1211sneqd 4032 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  { n }  =  { N } )
1310, 12fveq12d 5863 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =ring )  ->  ( (RSpan `  z ) `  {
n } )  =  ( S `  { N } ) )
147, 13oveq12d 6293 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z ~QG  ( (RSpan `  z ) `  { n } ) )  =  (ring ~QG  ( S `  { N } ) ) )
157, 14oveq12d 6293 . . . . . . . 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 2519 . . . . . . 7  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  =  U )
186, 17sylan9eqr 2523 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  s  =  U )
19 fvex 5867 . . . . . . . . . 10  |-  ( ZRHom `  s )  e.  _V
2019resex 5308 . . . . . . . . 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 5861 . . . . . . . . . . . . . 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 2462 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( n  =  0  <->  N  = 
0 ) )
2624oveq2d 6291 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( 0..^ n )  =  ( 0..^ N ) )
2725, 26ifbieq2d 3957 . . . . . . . . . . . . . . 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 2519 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  if (
n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  W )
3023, 29reseq12d 5265 . . . . . . . . . . . . 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 2519 . . . . . . . . . . . 12  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  F )
3322, 32sylan9eqr 2523 . . . . . . . . . . 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 5155 . . . . . . . . . 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 5169 . . . . . . . . . 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 5158 . . . . . . . . 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 2519 . . . . . . . 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 3455 . . . . . . 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 4213 . . . . . 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 6293 . . . . 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 3455 . . . 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 3455 . . 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 18304 . . 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 6300 . . 3  |-  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
4643, 44, 45fvmpt 5941 . 2  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
471, 46syl5eq 2513 1  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   [_csb 3428   ifcif 3932   {csn 4020   <.cop 4026   `'ccnv 4991    |` cres 4994    o. ccom 4996   ` cfv 5579  (class class class)co 6275   0cc0 9481    <_ cle 9618   NN0cn0 10784   ZZcz 10853  ..^cfzo 11781   ndxcnx 14476   sSet csts 14477   lecple 14551    /.s cqus 14749   ~QG cqg 15985   Ringcrg 16979  RSpancrsp 17593  ℤringzring 18249   ZRHomczrh 18297  ℤ/nczn 18300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-subg 15986  df-cmn 16589  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-subrg 17203  df-cnfld 18185  df-zring 18250  df-zn 18304
This theorem is referenced by:  znle  18333  znval2  18336
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