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Theorem lnr2i 26468
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 2316 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 lnr2i.u . . . . . 6  |-  U  =  (LIdeal `  R )
3 lnr2i.n . . . . . 6  |-  N  =  (RSpan `  R )
41, 2, 3islnr2 26466 . . . . 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 2322 . . . . . 6  |-  ( i  =  I  ->  (
i  =  ( N `
 g )  <->  I  =  ( N `  g ) ) )
76rexbidv 2598 . . . . 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 2916 . . . 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 26464 . . . . . . . . . . . 12  |-  ( R  e. LNoeR  ->  R  e.  Ring )
123, 1rspssid 16024 . . . . . . . . . . . 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 2825 . . . . . . . . . . 11  |-  g  e. 
_V
1615elpw 3665 . . . . . . . . . 10  |-  ( g  e.  ~P ( Base `  R )  <->  g  C_  ( Base `  R )
)
1715elpw 3665 . . . . . . . . . 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 3392 . . . . . . . 8  |-  ( g  e.  ( ~P ( Base `  R )  i^i 
Fin )  <->  ( g  e.  ~P ( Base `  R
)  /\  g  e.  Fin ) )
21 elin 3392 . . . . . . . 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 3662 . . . . . . . . . 10  |-  ( I  =  ( N `  g )  ->  ~P I  =  ~P ( N `  g )
)
2423ineq1d 3403 . . . . . . . . 9  |-  ( I  =  ( N `  g )  ->  ( ~P I  i^i  Fin )  =  ( ~P ( N `  g )  i^i  Fin ) )
2524eleq2d 2383 . . . . . . . 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 2686 . . 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 1633    e. wcel 1701   A.wral 2577   E.wrex 2578    i^i cin 3185    C_ wss 3186   ~Pcpw 3659   ` cfv 5292   Fincfn 6906   Basecbs 13195   Ringcrg 15386  LIdealclidl 15972  RSpancrsp 15973  LNoeRclnr 26461
This theorem is referenced by:  hbtlem6  26481
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-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
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-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  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-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-mgp 15375  df-rng 15389  df-ur 15391  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-lfig 26314  df-lnm 26322  df-lnr 26462
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