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Theorem znval 18699
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 zringring 18618 . . . . 5  |-ring  e.  Ring
32a1i 11 . . . 4  |-  ( n  =  N  ->ring  e.  Ring )
4 ovex 6324 . . . . . 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 5876 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  (RSpan ` ring ) )
9 znval.s . . . . . . . . . . . 12  |-  S  =  (RSpan ` ring )
108, 9syl6eqr 2516 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  S )
11 simpl 457 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  n  =  N )
1211sneqd 4044 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  { n }  =  { N } )
1310, 12fveq12d 5878 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =ring )  ->  ( (RSpan `  z ) `  {
n } )  =  ( S `  { N } ) )
147, 13oveq12d 6314 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z ~QG  ( (RSpan `  z ) `  { n } ) )  =  (ring ~QG  ( S `  { N } ) ) )
157, 14oveq12d 6314 . . . . . . . 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 2516 . . . . . . 7  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  =  U )
186, 17sylan9eqr 2520 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  s  =  U )
19 fvex 5882 . . . . . . . . . 10  |-  ( ZRHom `  s )  e.  _V
2019resex 5327 . . . . . . . . 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 5876 . . . . . . . . . . . . . 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 2459 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( n  =  0  <->  N  = 
0 ) )
2624oveq2d 6312 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( 0..^ n )  =  ( 0..^ N ) )
2725, 26ifbieq2d 3969 . . . . . . . . . . . . . . 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 2516 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  if (
n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  W )
3023, 29reseq12d 5284 . . . . . . . . . . . . 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 2516 . . . . . . . . . . . 12  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  F )
3322, 32sylan9eqr 2520 . . . . . . . . . . 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 5174 . . . . . . . . . 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 5188 . . . . . . . . . 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 5177 . . . . . . . . 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 2516 . . . . . . . 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 3457 . . . . . . 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 4226 . . . . . 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 6314 . . . . 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 3457 . . . 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 3457 . . 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 18671 . . 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 6324 . . 3  |-  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
4643, 44, 45fvmpt 5956 . 2  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
471, 46syl5eq 2510 1  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   [_csb 3430   ifcif 3944   {csn 4032   <.cop 4038   `'ccnv 5007    |` cres 5010    o. ccom 5012   ` cfv 5594  (class class class)co 6296   0cc0 9509    <_ cle 9646   NN0cn0 10816   ZZcz 10885  ..^cfzo 11821   ndxcnx 14641   sSet csts 14642   lecple 14719    /.s cqus 14922   ~QG cqg 16324   Ringcrg 17325  RSpancrsp 17944  ℤringzring 18615   ZRHomczrh 18664  ℤ/nczn 18667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-subg 16325  df-cmn 16927  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-subrg 17554  df-cnfld 18548  df-zring 18616  df-zn 18671
This theorem is referenced by:  znle  18700  znval2  18703
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