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Theorem lnr2i 29475
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 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 lnr2i.u . . . . . 6  |-  U  =  (LIdeal `  R )
3 lnr2i.n . . . . . 6  |-  N  =  (RSpan `  R )
41, 2, 3islnr2 29473 . . . . 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 2449 . . . . . 6  |-  ( i  =  I  ->  (
i  =  ( N `
 g )  <->  I  =  ( N `  g ) ) )
76rexbidv 2739 . . . . 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 3074 . . . 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 29471 . . . . . . . . . . . 12  |-  ( R  e. LNoeR  ->  R  e.  Ring )
123, 1rspssid 17308 . . . . . . . . . . . 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 2978 . . . . . . . . . . 11  |-  g  e. 
_V
1615elpw 3869 . . . . . . . . . 10  |-  ( g  e.  ~P ( Base `  R )  <->  g  C_  ( Base `  R )
)
1715elpw 3869 . . . . . . . . . 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 3542 . . . . . . . 8  |-  ( g  e.  ( ~P ( Base `  R )  i^i 
Fin )  <->  ( g  e.  ~P ( Base `  R
)  /\  g  e.  Fin ) )
21 elin 3542 . . . . . . . 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 3866 . . . . . . . . . 10  |-  ( I  =  ( N `  g )  ->  ~P I  =  ~P ( N `  g )
)
2423ineq1d 3554 . . . . . . . . 9  |-  ( I  =  ( N `  g )  ->  ( ~P I  i^i  Fin )  =  ( ~P ( N `  g )  i^i  Fin ) )
2524eleq2d 2510 . . . . . . . 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 2828 . . 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 1369    e. wcel 1756   A.wral 2718   E.wrex 2719    i^i cin 3330    C_ wss 3331   ~Pcpw 3863   ` cfv 5421   Fincfn 7313   Basecbs 14177   Ringcrg 16648  LIdealclidl 17254  RSpancrsp 17255  LNoeRclnr 29468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-sca 14257  df-vsca 14258  df-ip 14259  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-sbg 15550  df-subg 15681  df-mgp 16595  df-ur 16607  df-rng 16650  df-subrg 16866  df-lmod 16953  df-lss 17017  df-lsp 17056  df-sra 17256  df-rgmod 17257  df-lidl 17258  df-rsp 17259  df-lfig 29424  df-lnm 29432  df-lnr 29469
This theorem is referenced by:  hbtlem6  29488
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