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Theorem znzrhfo 16833
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 2439 . . . 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 16757 . . . . 5  |-  ZZ  e.  (SubRing ` fld )
3 eqid 2438 . . . . . 6  |-  (flds  ZZ )  =  (flds  ZZ )
43subrgbas 15882 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
52, 4mp1i 12 . . . 4  |-  ( N  e.  NN0  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
6 eqid 2438 . . . 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 6109 . . . . 5  |-  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  e.  _V
87a1i 11 . . . 4  |-  ( N  e.  NN0  ->  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  e.  _V )
9 ovex 6109 . . . . 5  |-  (flds  ZZ )  e.  _V
109a1i 11 . . . 4  |-  ( N  e.  NN0  ->  (flds  ZZ )  e.  _V )
111, 5, 6, 8, 10divslem 13773 . . 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 2438 . . . . . 6  |-  (RSpan `  (flds  ZZ ) )  =  (RSpan `  (flds  ZZ ) )
13 znzrhfo.y . . . . . 6  |-  Y  =  (ℤ/n `  N )
14 eqid 2438 . . . . . 6  |-  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  =  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) )
153, 12, 13, 14znbas 16829 . . . . 5  |-  ( N  e.  NN0  ->  ( ZZ
/. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  =  ( Base `  Y
) )
16 znzrhfo.b . . . . 5  |-  B  =  ( Base `  Y
)
1715, 16syl6eqr 2488 . . . 4  |-  ( N  e.  NN0  ->  ( ZZ
/. ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )  =  B )
18 foeq3 5654 . . . 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 16 . . 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 203 . 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 16831 . . 3  |-  ( N  e.  NN0  ->  L  =  ( x  e.  ZZ  |->  [ x ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
23 foeq1 5652 . . 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 16 . 2  |-  ( N  e.  NN0  ->  ( L : ZZ -onto-> B  <->  ( x  e.  ZZ  |->  [ x ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) : ZZ -onto-> B ) )
2520, 24mpbird 225 1  |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    e. cmpt 4269   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084   [cec 6906   /.cqs 6907   NN0cn0 10226   ZZcz 10287   Basecbs 13474   ↾s cress 13475    /.s cqus 13736   ~QG cqg 14945  SubRingcsubrg 15869  RSpancrsp 16248  ℂfldccnfld 16708   ZRHomczrh 16783  ℤ/nczn 16786
This theorem is referenced by:  zncyg  16834  znf1o  16837  zzngim  16838  znfld  16846  znunit  16849  znrrg  16851  cygznlem2a  16853  cygznlem3  16855  dchrelbas4  21032  dchrzrhcl  21034  lgsdchrval  21136  lgsdchr  21137  rpvmasumlem  21186  dchrmusum2  21193  dchrvmasumlem3  21198  dchrisum0ff  21206  dchrisum0flblem1  21207  rpvmasum2  21211  dchrisum0re  21212  dchrisum0lem2a  21216  dirith  21228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-seq 11329  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-0g 13732  df-imas 13739  df-divs 13740  df-mnd 14695  df-mhm 14743  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-nsg 14947  df-eqg 14948  df-ghm 15009  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-oppr 15733  df-rnghom 15824  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-sra 16249  df-rgmod 16250  df-lidl 16251  df-rsp 16252  df-2idl 16308  df-cnfld 16709  df-zrh 16787  df-zn 16790
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