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Theorem znzrhfo 16557
Description: The  ZZ ring homomorphism is a surjection onto  ZZ  /  n ZZ. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
znzrhfo.y  |-  Y  =  (ℤ/n `  N )
znzrhfo.b  |-  B  =  ( Base `  Y
)
znzrhfo.2  |-  L  =  ( ZRHom `  Y
)
Assertion
Ref Expression
znzrhfo  |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )

Proof of Theorem znzrhfo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqidd 2317 . . . 4  |-  ( N  e.  NN0  ->  ( (flds  ZZ ) 
/.s  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  =  ( (flds  ZZ )  /.s  ( (flds  ZZ ) ~QG  (
(RSpan `  (flds  ZZ ) ) `  { N } ) ) ) )
2 zsubrg 16481 . . . . 5  |-  ZZ  e.  (SubRing ` fld )
3 eqid 2316 . . . . . 6  |-  (flds  ZZ )  =  (flds  ZZ )
43subrgbas 15603 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
52, 4mp1i 11 . . . 4  |-  ( N  e.  NN0  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
6 eqid 2316 . . . 4  |-  ( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  =  ( x  e.  ZZ  |->  [ x ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )
7 ovex 5925 . . . . 5  |-  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  e.  _V
87a1i 10 . . . 4  |-  ( N  e.  NN0  ->  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  e.  _V )
9 ovex 5925 . . . . 5  |-  (flds  ZZ )  e.  _V
109a1i 10 . . . 4  |-  ( N  e.  NN0  ->  (flds  ZZ )  e.  _V )
111, 5, 6, 8, 10divslem 13494 . . 3  |-  ( N  e.  NN0  ->  ( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> ( ZZ
/. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
12 eqid 2316 . . . . . 6  |-  (RSpan `  (flds  ZZ ) )  =  (RSpan `  (flds  ZZ ) )
13 znzrhfo.y . . . . . 6  |-  Y  =  (ℤ/n `  N )
14 eqid 2316 . . . . . 6  |-  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  =  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) )
153, 12, 13, 14znbas 16553 . . . . 5  |-  ( N  e.  NN0  ->  ( ZZ
/. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  =  ( Base `  Y
) )
16 znzrhfo.b . . . . 5  |-  B  =  ( Base `  Y
)
1715, 16syl6eqr 2366 . . . 4  |-  ( N  e.  NN0  ->  ( ZZ
/. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  =  B )
18 foeq3 5487 . . . 4  |-  ( ( ZZ /. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) ) )  =  B  -> 
( ( x  e.  ZZ  |->  [ x ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> ( ZZ
/. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  <-> 
( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> B ) )
1917, 18syl 15 . . 3  |-  ( N  e.  NN0  ->  ( ( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) ) ) : ZZ -onto-> ( ZZ /. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  <-> 
( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> B ) )
2011, 19mpbid 201 . 2  |-  ( N  e.  NN0  ->  ( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> B )
21 znzrhfo.2 . . . 4  |-  L  =  ( ZRHom `  Y
)
223, 12, 14, 13, 21znzrh2 16555 . . 3  |-  ( N  e.  NN0  ->  L  =  ( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
23 foeq1 5485 . . 3  |-  ( L  =  ( x  e.  ZZ  |->  [ x ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  ->  ( L : ZZ -onto-> B  <->  ( x  e.  ZZ  |->  [ x ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> B ) )
2422, 23syl 15 . 2  |-  ( N  e.  NN0  ->  ( L : ZZ -onto-> B  <->  ( x  e.  ZZ  |->  [ x ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> B ) )
2520, 24mpbird 223 1  |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   _Vcvv 2822   {csn 3674    e. cmpt 4114   -onto->wfo 5290   ` cfv 5292  (class class class)co 5900   [cec 6700   /.cqs 6701   NN0cn0 10012   ZZcz 10071   Basecbs 13195   ↾s cress 13196    /.s cqus 13457   ~QG cqg 14666  SubRingcsubrg 15590  RSpancrsp 15973  ℂfldccnfld 16432   ZRHomczrh 16507  ℤ/nczn 16510
This theorem is referenced by:  zncyg  16558  znf1o  16561  zzngim  16562  znfld  16570  znunit  16573  znrrg  16575  cygznlem2a  16577  cygznlem3  16579  dchrelbas4  20535  dchrzrhcl  20537  lgsdchrval  20639  lgsdchr  20640  rpvmasumlem  20689  dchrmusum2  20696  dchrvmasumlem3  20701  dchrisum0ff  20709  dchrisum0flblem1  20710  rpvmasum2  20714  dchrisum0re  20715  dchrisum0lem2a  20719  dirith  20731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-addf 8861  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-ec 6704  df-qs 6708  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-seq 11094  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-0g 13453  df-imas 13460  df-divs 13461  df-mnd 14416  df-mhm 14464  df-grp 14538  df-minusg 14539  df-sbg 14540  df-mulg 14541  df-subg 14667  df-nsg 14668  df-eqg 14669  df-ghm 14730  df-cmn 15140  df-abl 15141  df-mgp 15375  df-rng 15389  df-cring 15390  df-ur 15391  df-oppr 15454  df-rnghom 15545  df-subrg 15592  df-lmod 15678  df-lss 15739  df-lsp 15778  df-sra 15974  df-rgmod 15975  df-lidl 15976  df-rsp 15977  df-2idl 16033  df-cnfld 16433  df-zrh 16511  df-zn 16514
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