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Theorem lnr2i 27320
Description: Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
lnr2i.u  |-  U  =  (LIdeal `  R )
lnr2i.n  |-  N  =  (RSpan `  R )
Assertion
Ref Expression
lnr2i  |-  ( ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P I  i^i 
Fin ) I  =  ( N `  g
) )
Distinct variable groups:    g, I    g, N    R, g    U, g

Proof of Theorem lnr2i
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 lnr2i.u . . . . . 6  |-  U  =  (LIdeal `  R )
3 lnr2i.n . . . . . 6  |-  N  =  (RSpan `  R )
41, 2, 3islnr2 27318 . . . . 5  |-  ( R  e. LNoeR 
<->  ( R  e.  Ring  /\ 
A. i  e.  U  E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
i  =  ( N `
 g ) ) )
54simprbi 450 . . . 4  |-  ( R  e. LNoeR  ->  A. i  e.  U  E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
i  =  ( N `
 g ) )
6 eqeq1 2289 . . . . . 6  |-  ( i  =  I  ->  (
i  =  ( N `
 g )  <->  I  =  ( N `  g ) ) )
76rexbidv 2564 . . . . 5  |-  ( i  =  I  ->  ( E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
i  =  ( N `
 g )  <->  E. g  e.  ( ~P ( Base `  R )  i^i  Fin ) I  =  ( N `  g )
) )
87rspcva 2882 . . . 4  |-  ( ( I  e.  U  /\  A. i  e.  U  E. g  e.  ( ~P ( Base `  R )  i^i  Fin ) i  =  ( N `  g
) )  ->  E. g  e.  ( ~P ( Base `  R )  i^i  Fin ) I  =  ( N `  g )
)
95, 8sylan2 460 . . 3  |-  ( ( I  e.  U  /\  R  e. LNoeR )  ->  E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
I  =  ( N `
 g ) )
109ancoms 439 . 2  |-  ( ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P ( Base `  R )  i^i  Fin ) I  =  ( N `  g )
)
11 lnrrng 27316 . . . . . . . . . . . 12  |-  ( R  e. LNoeR  ->  R  e.  Ring )
123, 1rspssid 15975 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  g  C_  ( Base `  R
) )  ->  g  C_  ( N `  g
) )
1311, 12sylan 457 . . . . . . . . . . 11  |-  ( ( R  e. LNoeR  /\  g  C_  ( Base `  R
) )  ->  g  C_  ( N `  g
) )
1413ex 423 . . . . . . . . . 10  |-  ( R  e. LNoeR  ->  ( g  C_  ( Base `  R )  ->  g  C_  ( N `  g ) ) )
15 vex 2791 . . . . . . . . . . 11  |-  g  e. 
_V
1615elpw 3631 . . . . . . . . . 10  |-  ( g  e.  ~P ( Base `  R )  <->  g  C_  ( Base `  R )
)
1715elpw 3631 . . . . . . . . . 10  |-  ( g  e.  ~P ( N `
 g )  <->  g  C_  ( N `  g ) )
1814, 16, 173imtr4g 261 . . . . . . . . 9  |-  ( R  e. LNoeR  ->  ( g  e. 
~P ( Base `  R
)  ->  g  e.  ~P ( N `  g
) ) )
1918anim1d 547 . . . . . . . 8  |-  ( R  e. LNoeR  ->  ( ( g  e.  ~P ( Base `  R )  /\  g  e.  Fin )  ->  (
g  e.  ~P ( N `  g )  /\  g  e.  Fin ) ) )
20 elin 3358 . . . . . . . 8  |-  ( g  e.  ( ~P ( Base `  R )  i^i 
Fin )  <->  ( g  e.  ~P ( Base `  R
)  /\  g  e.  Fin ) )
21 elin 3358 . . . . . . . 8  |-  ( g  e.  ( ~P ( N `  g )  i^i  Fin )  <->  ( g  e.  ~P ( N `  g )  /\  g  e.  Fin ) )
2219, 20, 213imtr4g 261 . . . . . . 7  |-  ( R  e. LNoeR  ->  ( g  e.  ( ~P ( Base `  R )  i^i  Fin )  ->  g  e.  ( ~P ( N `  g )  i^i  Fin ) ) )
23 pweq 3628 . . . . . . . . . 10  |-  ( I  =  ( N `  g )  ->  ~P I  =  ~P ( N `  g )
)
2423ineq1d 3369 . . . . . . . . 9  |-  ( I  =  ( N `  g )  ->  ( ~P I  i^i  Fin )  =  ( ~P ( N `  g )  i^i  Fin ) )
2524eleq2d 2350 . . . . . . . 8  |-  ( I  =  ( N `  g )  ->  (
g  e.  ( ~P I  i^i  Fin )  <->  g  e.  ( ~P ( N `  g )  i^i  Fin ) ) )
2625imbi2d 307 . . . . . . 7  |-  ( I  =  ( N `  g )  ->  (
( g  e.  ( ~P ( Base `  R
)  i^i  Fin )  ->  g  e.  ( ~P I  i^i  Fin )
)  <->  ( g  e.  ( ~P ( Base `  R )  i^i  Fin )  ->  g  e.  ( ~P ( N `  g )  i^i  Fin ) ) ) )
2722, 26syl5ibrcom 213 . . . . . 6  |-  ( R  e. LNoeR  ->  ( I  =  ( N `  g
)  ->  ( g  e.  ( ~P ( Base `  R )  i^i  Fin )  ->  g  e.  ( ~P I  i^i  Fin ) ) ) )
2827imdistand 673 . . . . 5  |-  ( R  e. LNoeR  ->  ( ( I  =  ( N `  g )  /\  g  e.  ( ~P ( Base `  R )  i^i  Fin ) )  ->  (
I  =  ( N `
 g )  /\  g  e.  ( ~P I  i^i  Fin ) ) ) )
29 ancom 437 . . . . 5  |-  ( ( g  e.  ( ~P ( Base `  R
)  i^i  Fin )  /\  I  =  ( N `  g )
)  <->  ( I  =  ( N `  g
)  /\  g  e.  ( ~P ( Base `  R
)  i^i  Fin )
) )
30 ancom 437 . . . . 5  |-  ( ( g  e.  ( ~P I  i^i  Fin )  /\  I  =  ( N `  g )
)  <->  ( I  =  ( N `  g
)  /\  g  e.  ( ~P I  i^i  Fin ) ) )
3128, 29, 303imtr4g 261 . . . 4  |-  ( R  e. LNoeR  ->  ( ( g  e.  ( ~P ( Base `  R )  i^i 
Fin )  /\  I  =  ( N `  g ) )  -> 
( g  e.  ( ~P I  i^i  Fin )  /\  I  =  ( N `  g ) ) ) )
3231reximdv2 2652 . . 3  |-  ( R  e. LNoeR  ->  ( E. g  e.  ( ~P ( Base `  R )  i^i  Fin ) I  =  ( N `  g )  ->  E. g  e.  ( ~P I  i^i  Fin ) I  =  ( N `  g )
) )
3332adantr 451 . 2  |-  ( ( R  e. LNoeR  /\  I  e.  U )  ->  ( E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
I  =  ( N `
 g )  ->  E. g  e.  ( ~P I  i^i  Fin )
I  =  ( N `
 g ) ) )
3410, 33mpd 14 1  |-  ( ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P I  i^i 
Fin ) I  =  ( N `  g
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255   Fincfn 6863   Basecbs 13148   Ringcrg 15337  LIdealclidl 15923  RSpancrsp 15924  LNoeRclnr 27313
This theorem is referenced by:  hbtlem6  27333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-lfig 27166  df-lnm 27174  df-lnr 27314
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