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Theorem lnr2i 30669
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 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 lnr2i.u . . . . . 6  |-  U  =  (LIdeal `  R )
3 lnr2i.n . . . . . 6  |-  N  =  (RSpan `  R )
41, 2, 3islnr2 30667 . . . . 5  |-  ( R  e. LNoeR 
<->  ( R  e.  Ring  /\ 
A. i  e.  U  E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
i  =  ( N `
 g ) ) )
54simprbi 464 . . . 4  |-  ( R  e. LNoeR  ->  A. i  e.  U  E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
i  =  ( N `
 g ) )
6 eqeq1 2471 . . . . . 6  |-  ( i  =  I  ->  (
i  =  ( N `
 g )  <->  I  =  ( N `  g ) ) )
76rexbidv 2973 . . . . 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 3212 . . . 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 474 . . 3  |-  ( ( I  e.  U  /\  R  e. LNoeR )  ->  E. g  e.  ( ~P ( Base `  R
)  i^i  Fin )
I  =  ( N `
 g ) )
109ancoms 453 . 2  |-  ( ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P ( Base `  R )  i^i  Fin ) I  =  ( N `  g )
)
11 lnrrng 30665 . . . . . . . . . . . 12  |-  ( R  e. LNoeR  ->  R  e.  Ring )
123, 1rspssid 17653 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  g  C_  ( Base `  R
) )  ->  g  C_  ( N `  g
) )
1311, 12sylan 471 . . . . . . . . . . 11  |-  ( ( R  e. LNoeR  /\  g  C_  ( Base `  R
) )  ->  g  C_  ( N `  g
) )
1413ex 434 . . . . . . . . . 10  |-  ( R  e. LNoeR  ->  ( g  C_  ( Base `  R )  ->  g  C_  ( N `  g ) ) )
15 vex 3116 . . . . . . . . . . 11  |-  g  e. 
_V
1615elpw 4016 . . . . . . . . . 10  |-  ( g  e.  ~P ( Base `  R )  <->  g  C_  ( Base `  R )
)
1715elpw 4016 . . . . . . . . . 10  |-  ( g  e.  ~P ( N `
 g )  <->  g  C_  ( N `  g ) )
1814, 16, 173imtr4g 270 . . . . . . . . 9  |-  ( R  e. LNoeR  ->  ( g  e. 
~P ( Base `  R
)  ->  g  e.  ~P ( N `  g
) ) )
1918anim1d 564 . . . . . . . 8  |-  ( R  e. LNoeR  ->  ( ( g  e.  ~P ( Base `  R )  /\  g  e.  Fin )  ->  (
g  e.  ~P ( N `  g )  /\  g  e.  Fin ) ) )
20 elin 3687 . . . . . . . 8  |-  ( g  e.  ( ~P ( Base `  R )  i^i 
Fin )  <->  ( g  e.  ~P ( Base `  R
)  /\  g  e.  Fin ) )
21 elin 3687 . . . . . . . 8  |-  ( g  e.  ( ~P ( N `  g )  i^i  Fin )  <->  ( g  e.  ~P ( N `  g )  /\  g  e.  Fin ) )
2219, 20, 213imtr4g 270 . . . . . . 7  |-  ( R  e. LNoeR  ->  ( g  e.  ( ~P ( Base `  R )  i^i  Fin )  ->  g  e.  ( ~P ( N `  g )  i^i  Fin ) ) )
23 pweq 4013 . . . . . . . . . 10  |-  ( I  =  ( N `  g )  ->  ~P I  =  ~P ( N `  g )
)
2423ineq1d 3699 . . . . . . . . 9  |-  ( I  =  ( N `  g )  ->  ( ~P I  i^i  Fin )  =  ( ~P ( N `  g )  i^i  Fin ) )
2524eleq2d 2537 . . . . . . . 8  |-  ( I  =  ( N `  g )  ->  (
g  e.  ( ~P I  i^i  Fin )  <->  g  e.  ( ~P ( N `  g )  i^i  Fin ) ) )
2625imbi2d 316 . . . . . . 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 222 . . . . . 6  |-  ( R  e. LNoeR  ->  ( I  =  ( N `  g
)  ->  ( g  e.  ( ~P ( Base `  R )  i^i  Fin )  ->  g  e.  ( ~P I  i^i  Fin ) ) ) )
2827imdistand 692 . . . . 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 450 . . . . 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 450 . . . . 5  |-  ( ( g  e.  ( ~P I  i^i  Fin )  /\  I  =  ( N `  g )
)  <->  ( I  =  ( N `  g
)  /\  g  e.  ( ~P I  i^i  Fin ) ) )
3128, 29, 303imtr4g 270 . . . 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 2934 . . 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 465 . 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 15 1  |-  ( ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P I  i^i 
Fin ) I  =  ( N `  g
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   ` cfv 5586   Fincfn 7513   Basecbs 14486   Ringcrg 16986  LIdealclidl 17599  RSpancrsp 17600  LNoeRclnr 30662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-mgp 16932  df-ur 16944  df-rng 16988  df-subrg 17210  df-lmod 17297  df-lss 17362  df-lsp 17401  df-sra 17601  df-rgmod 17602  df-lidl 17603  df-rsp 17604  df-lfig 30618  df-lnm 30626  df-lnr 30663
This theorem is referenced by:  hbtlem6  30682
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