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Theorem znval 19104
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 19040 . . . . 5  |-ring  e.  Ring
32a1i 11 . . . 4  |-  ( n  =  N  ->ring  e.  Ring )
4 ovex 6333 . . . . . 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 462 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =ring )  ->  z  =ring )
87fveq2d 5885 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  (RSpan ` ring ) )
9 znval.s . . . . . . . . . . . 12  |-  S  =  (RSpan ` ring )
108, 9syl6eqr 2481 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  (RSpan `  z )  =  S )
11 simpl 458 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  z  =ring )  ->  n  =  N )
1211sneqd 4010 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  z  =ring )  ->  { n }  =  { N } )
1310, 12fveq12d 5887 . . . . . . . . . 10  |-  ( ( n  =  N  /\  z  =ring )  ->  ( (RSpan `  z ) `  {
n } )  =  ( S `  { N } ) )
147, 13oveq12d 6323 . . . . . . . . 9  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z ~QG  ( (RSpan `  z ) `  { n } ) )  =  (ring ~QG  ( S `  { N } ) ) )
157, 14oveq12d 6323 . . . . . . . 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 2481 . . . . . . 7  |-  ( ( n  =  N  /\  z  =ring )  ->  ( z 
/.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) )  =  U )
186, 17sylan9eqr 2485 . . . . . 6  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  s  =  U )
19 fvex 5891 . . . . . . . . . 10  |-  ( ZRHom `  s )  e.  _V
2019resex 5167 . . . . . . . . 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 5885 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ZRHom `  s )  =  ( ZRHom `  U )
)
24 simpll 758 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  n  =  N )
2524eqeq1d 2424 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( n  =  0  <->  N  = 
0 ) )
2624oveq2d 6321 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( 0..^ n )  =  ( 0..^ N ) )
2725, 26ifbieq2d 3936 . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . 14  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  if (
n  =  0 ,  ZZ ,  ( 0..^ n ) )  =  W )
3023, 29reseq12d 5125 . . . . . . . . . . . . 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 2481 . . . . . . . . . . . 12  |-  ( ( ( n  =  N  /\  z  =ring )  /\  s  =  ( z  /.s  ( z ~QG  ( (RSpan `  z
) `  { n } ) ) ) )  ->  ( ( ZRHom `  s )  |`  if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  =  F )
3322, 32sylan9eqr 2485 . . . . . . . . . . 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 5015 . . . . . . . . . 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 5029 . . . . . . . . . 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 5018 . . . . . . . . 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 2481 . . . . . . . 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 3422 . . . . . . 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 4194 . . . . . 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 6323 . . . . 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 3422 . . . 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 3422 . . 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 19076 . . 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 6333 . . 3  |-  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
4643, 44, 45fvmpt 5964 . 2  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  =  ( U sSet  <. ( le `  ndx ) ,  .<_  >. )
)
471, 46syl5eq 2475 1  |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <. ( le `  ndx ) , 
.<_  >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080   [_csb 3395   ifcif 3911   {csn 3998   <.cop 4004   `'ccnv 4852    |` cres 4855    o. ccom 4857   ` cfv 5601  (class class class)co 6305   0cc0 9546    <_ cle 9683   NN0cn0 10876   ZZcz 10944  ..^cfzo 11922   ndxcnx 15117   sSet csts 15118   lecple 15196    /.s cqus 15403   ~QG cqg 16812   Ringcrg 17779  RSpancrsp 18393  ℤringzring 19037   ZRHomczrh 19069  ℤ/nczn 19072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-addf 9625  ax-mulf 9626
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-starv 15204  df-tset 15208  df-ple 15209  df-ds 15211  df-unif 15212  df-0g 15339  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-grp 16672  df-minusg 16673  df-subg 16813  df-cmn 17431  df-mgp 17723  df-ur 17735  df-ring 17781  df-cring 17782  df-subrg 18005  df-cnfld 18970  df-zring 19038  df-zn 19076
This theorem is referenced by:  znle  19105  znval2  19106
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